              This is the final preprint version of a paper which appeared in
  Publicacions Matemàticas 59 (2015), 271--300.  The published version is accessible at
                       http://dx.doi.org/10.5565/PUBLMAT_59215_01 .

MINIMAL FAITHFUL MODULES OVER ARTINIAN RINGS

GEORGE M. BERGMAN

Abstract. Let R be a left Artinian ring, and M a faithful left R-module such that no proper submodule or homomorphic image of M is faithful.

If R is local, and socle(R) is central in R, we show that length(M/J(R)M) + length(socle(M)) ≤ length(socle(R)) + 1.

If R is a finite-dimensional algebra over an algebraically closed field, but not necessarily local or having central socle, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and the summand +1 replaced by the Euler characteristic of a graph determined by the bimodule structure of socle(R). The statement proved is slightly more general than this summary; we examine the question of whether much stronger generalizations are possible.

If a faithful module M over an Artinian ring R is only assumed to have one of the above minimality properties – no faithful proper submodules, or no faithful proper homomorphic images – we find that the length of M/J(R)M in the former case, and of socle(M) in the latter, is ≤ length(socle(R)). The proofs involve general lemmas on decompositions of modules.

1. Background and motivation

This paper arose as a tangent to the unpublished note [1], which examines the question of which commutative Artinian rings R have the property that every faithful R-module M has length greater than or equal to that of R. (If R is a commutative algebra over a field k, it is known that this is true if R can be generated over k by 2 elements, but false for 4-generator algebras; it is an open problem whether it holds for 3-generator algebras. For more on this, see [1] and [5, Chapter 5].)

In studying that question, it is natural to focus on faithful modules M no proper factor-modules or submodules of which are faithful. I obtained some results showing that such M must have small “top” M/J(R)M and “bottom” socle(M); Luchezar Avramov then pointed me to a 1972 paper of Tor Gulliksen, [3], which obtained a stronger result; I found, in turn, that Gulliksen’s bound could be strengthened, and that the strengthened result could be applied to a wider class of rings (which, in particular, need not be commutative). This will be done in §2 below.

Some details: the relevant result of Gulliksen’s paper is that if R is a commutative local Artinian ring, and M a faithful R-module no proper submodule or homomorphic image of which is faithful, then each of the semisimple R-modules M/J(R)M and socle(M) has length less than or equal to that of socle(R), with at least one of these inequalities strict unless M ≅ R. His proof is, in effect, a lemma in linear algebra, about bilinear maps A × B → C of finite-dimensional vector spaces over a field, which have the property that every nonzero element of A acts nontrivially, but which lose this property both on restriction to any proper subspace of B, and on composition with any noninjective map out of C; though he only states it for the natural map socle(R) × M/J(R)M → socle(M) of vector spaces over the field R/J(R). In §2 we show that, in the general linear algebra setting, one has the stronger inequality dim(A) ≥ dim(B) + dim(C) − 1. We then, like Gulliksen, apply this result to maps socle(R) × M/J(R)M → socle(M); here, rather than assuming the local ring R commutative, it is only necessary to assume socle(R) central in R.

2010 Mathematics Subject Classification. Primary: 13E10, 16G10, 16P20, Secondary: 16D60.

Key words and phrases. faithful modules over Artinian rings; length of a module or bimodule; socle of a ring or module.

After publication of this note, updates, errata, related references etc., if found, will be recorded at http://math.berkeley.

edu/~gbergman/papers/ .

Subsequent sections obtain inequalities of a similar nature for modules over not necessarily local Artinian rings R.

In §6, which is essentially independent of the rest of this note, we obtain results on lengths of modules satisfying only one of our minimality conditions.

2. Faithful modules over local Artin rings with central socles

Before formulating the promised linear algebra result, let us note that for finite-dimensional vector spaces B and C over a field k, to give a linear map a : B → C is equivalent to giving an element of B^{∗} ⊗_{k} C. Hence a k-vector space A, given with a k-bilinear map A × B → C such that every nonzero element of A induces a nonzero map B → C is equivalent to a subspace A ⊆ B^{∗} ⊗_{k} C. This is a more symmetric situation; so we shall formulate the linear algebra result in that form, with B^{∗} re-named B.

As noted in the preceding section, the bilinear maps A×B → C of interest are those such that restriction to any proper subspace of B, or composition with the natural map into any proper homomorphic image of C, kills the action of some element of A. With B dualized as above, this becomes the condition that passing to a homomorphic image of either B or C has that effect. Now a minimal proper homomorphic image of B has the form B/kb for some b ∈ B − {0}, so the condition that passage to any such homomorphic image kills some element of A says that for each nonzero b ∈ B, the space A contains a nonzero element of the form b ⊗ c. Likewise, the condition that passing to any proper homomorphic image C/kc of C kills some element of A means that for each nonzero c ∈ C, the space A contains a nonzero element of the form b⊗c. This leads to the formulation of the next result.

My original proof required that the field k have cardinality at least max(dim_{k}(B), dim_{k}(C)). For the present proof, I am indebted to Clément de Seguins Pazzis [6].

Proposition 1. Let k be a field, and B and C nonzero finite-dimensional vector spaces over k. Suppose A ⊆ B ⊗_{k} C is a subspace such that

(1)  (∀ b ∈ B − {0}) (∃ c ∈ C − {0}) b ⊗ c ∈ A,

and

(2)  (∀ c ∈ C − {0}) (∃ b ∈ B − {0}) b ⊗ c ∈ A.

Then

(3)  dim_{k}(A) ≥ dim_{k}(B) + dim_{k}(C) − 1.

Proof. (After de Seguins Pazzis [6].)

If dim_{k}(B) or dim_{k}(C) is 1, then (2), respectively (1), says that A = B ⊗ C, and the desired result is immediate. So let m, n > 1, assume inductively that the result is known when B has dimension m − 1 and C has dimension n − 1, and suppose we are in a situation with dim_{k}(B) = m and dim_{k}(C) = n. We consider two cases:

Case 1. There exists a linear functional f : B → k such that the induced map f⊗ id_{C} : B⊗C → k⊗C ≅ C carries A ⊆ B ⊗ C surjectively onto C.

Then taking any basis {c_{1}, …, c_{n}} of C, we can find elements a_{1}, …, a_{n} ∈ A whose images under f ⊗ id_{C} are c_{1}, …, c_{n}. On the other hand, letting b_{1}, …, b_{m−1} be any basis of ker(f) ⊆ B, we can find, by (1), nonzero c′_{1}, …, c′_{m−1} ∈ C such that b_{1} ⊗ c′_{1}, …, b_{m−1} ⊗ c′_{m−1} ∈ A. We claim that a_{1}, …, a_{n}, b_{1} ⊗ c′_{1}, …, b_{m−1} ⊗c′_{m−1} ∈ A are linearly independent. Indeed, given any linear dependence relation among these elements, if we apply f ⊗ id_{C} to it, this will annihilate the last m − 1 terms, and looking at the remaining terms, we conclude from the linear independence of c_{1}, …, c_{n} that the coefficients of a_{1}, …, a_{n} in the relation must be zero. Given this fact, if for any i ≤ m − 1 we let f_{i} : B → k be a linear functional which takes b_{i} to 1 and all other b_{j} to 0, then application of f_{i} ⊗ id_{C} to our relation shows that the coefficient of b_{i} ⊗ c′_{i} is also zero. So the indicated m + n − 1 elements of A are linearly independent, establishing (3) in this case.

The negation of the condition of Case 1 says that for every linear functional f : B → k, a certain conclusion holds. But to finish the proof, it will be enough to assume this for some nonzero f, as we do in

Case 2. There exists a nonzero linear functional f : B → k such that the induced map f ⊗ id_{C} : B ⊗C → k ⊗ C ≅ C carries A ⊆ B ⊗ C into a proper subspace C_{0} ⊆ C.

Without loss of generality, we can take C_{0} to have dimension n − 1. Let B_{0} = ker(f), which has dimension m − 1. Then our assumption on f implies

(4) For any element of A of the form b ⊗ c, either b ∈ B_{0} or c ∈ C_{0}.

Now let p_{B} be any retraction B → B_{0}, let p_{C} be any retraction C → C_{0}, and let A_{0} be the image of A under p_{B} ⊗ p_{C} : B ⊗ C → B_{0} ⊗ C_{0}. (It is not asserted that A_{0} ⊆ A.) I claim that the analogs of (1) and (2) hold with A_{0}, B_{0}, C_{0} in the roles of A, B and C. Indeed, given nonzero b ∈ B_{0}, let b′ ∈ B be chosen which projects to b under p_{B}, but is not in B_{0}; and use (1) to find a nonzero c ∈ C such that b′ ⊗ c ∈ A. Then by (4), c ∈ C_{0}, so c is fixed by p_{C}; so applying p_{B} ⊗ p_{C} to b′ ⊗ c, we get b ⊗ c ∈ A_{0}; the analog of (1). The symmetric argument gives the analog of (2).

Hence by our inductive assumption, dim(A_{0}) ≥ (m − 1) + (n − 1) − 1, which is short by 2 of the desired lower bound on the dimension of A. Since A_{0} is the image of A under p_{B} ⊗ p_{C}, it will suffice to find two linearly independent elements of A which are in the kernel of that map. To do so, let b span ker(p_{B}) and c span ker(p_{C}), and use (1) and (2) to find nonzero elements b ⊗ c′ and b′ ⊗ c of A. From (4), one sees that these are linearly independent, completing the proof. □

Dualizing B, to pass from this result to a statement about linear maps, as discussed earlier, we get

Corollary 2. Suppose A, B and C are finite-dimensional vector spaces over a field k, and f : A×B → C a bilinear map, such that every nonzero element of A induces a nonzero map B → C, but such that this property is lost on restriction to any proper subspace of B, and likewise on composition with the map to any proper homomorphic image of C. Then dim(A) ≥ dim(B) + dim(C) − 1. □

We shall now deduce the asserted generalization of Gulliksen’s result. (Incidentally, Gulliksen does not explicitly say in [3] that his rings are commutative; but this is apparent from the techniques he uses, e.g., the duality called on at the top of [3, p. 79]; and is also evidenced by the fact that commutativity is one of the properties he verifies for the matrix example of [3, Theorem 2]. In the present note, rings are not assumed commutative unless this is stated.)

Recall that the socle of a left or right module M is the sum of its simple submodules. By the left and right socles of a ring R, we understand socle(_{R}R) and socle(R_{R}), the socles of R as a left and as a right module, each of which is a 2-sided ideal of R. On the other hand, by the socle of R, socle(R), we shall mean the sum of all its minimal nonzero 2-sided ideals, which, for R left or right Artinian, is the intersection of its right and left socles. (So, for example, in the algebra of n × n upper triangular matrices over a field, the left socle is the ideal of matrices with support in the top row, the right socle is the ideal of matrices with support in the last column, and socle(R) is the ideal of matrices with support in the upper right-hand corner.)

We denote the Jacobson radical of a ring R by J(R).

Theorem 3 (cf. Gulliksen [3, Lemma 2]). Suppose R is a left Artinian local ring such that socle(R) is central in R, and let M be any faithful left R-module such that no proper submodule or homomorphic image of M is faithful. Then

(5)  dim_{R/J(R)}(M/J(R)M) + dim_{R/J(R)}(socle(M)) ≤ dim_{R/J(R)}(socle(R)) + 1.

Proof. J(R) annihilates M/J(R)M and socle(M) as left R-modules, and annihilates the (R, R)-bimodule socle(R) on both sides; so the first two become left vector spaces, and the latter a bimodule, over the division ring R/J(R). The statement that socle(R) is central in R says that R/J(R) has the same action on the two sides of socle(R), from which it immediately follows that the division ring R/J(R) must be a field k. Writing A = socle(R), B = M/J(R)M, C = socle(M), the left module operation of R on M induces an action by which A carries B into C, giving a k-bilinear map A × B → C. Because M is faithful, every nonzero element of A gives a nonzero map B → C, while the minimality assumptions on M imply that this property of our bilinear map is lost when one passes to any proper subspace of B or proper homomorphic image of C. (Indeed, if B_{0} is a proper k-subspace of B, equivalently, a proper R-submodule, then its inverse image in M is a proper submodule, hence non-faithful. The kernel of the action of R on this submodule must meet socle(R) = A, and any a ∈ A in that kernel acts trivially on B_{0}. The dual argument applies to proper homomorphic images C_{0} of C.) That B and C are finite-dimensional is also not hard to see from these minimality conditions. (Some general results of which this finite dimensionality is a special case are developed in §6.)

Hence Corollary 2 applies, and gives the desired inequality. □

Remarks: If R is a left Artinian local ring, then a sufficient condition for socle(R) to be central is that R be an algebra over a field k, and that the residue field of R be k itself. Such an R can be far from commutative; for instance, over any field we can take for R the ring of upper triangular n × n matrices over k with scalar main diagonal. On the other hand, if R contains a field k which is not central in R, but which again maps isomorphically onto R/J(R), then socle(R) may or may not be central. For example, if we take a twisted polynomial ring k[x; θ] where θ is an automorphism of k of finite order d > 1, and look at its local factor-ring k[x; θ]/(x^{n+1}) for some n > 0, then k is not central in R, but the socle, k x^{n}, is central if and only n is a multiple of d. Likewise, given a field k and automorphisms θ_{2}, …, θ_{n}, we can generalize the triangular-matrices example by letting R be the ring of upper-triangular n × n matrices ((a_{ij})) over k satisfying a_{ii} = θ_{i}(a_{11}) for 2 ≤ i ≤ n. This will be Artinian and local with residue field isomorphic to k, and its socle, k e_{1n}, will be central if and only if θ_{n} = id.

Turning back to Proposition 1, one may ask what a minimal subspace A ⊆ B ⊗ C satisfying (1) and (2) can look like. Easy examples are the spaces of the form B ⊗ c + b ⊗ C for arbitrary nonzero elements b ∈ B, c ∈ C. These spaces have dimension exactly dim(B) + dim(C) − 1, since B ⊗ c and b ⊗ C intersect in the one-dimensional subspace spanned by b ⊗ c. In matrix notation, this example can be pictured as the vector space of m × n matrices with support in the union of the first row and the first column.

Using such matrix notation, one can describe further minimal families. For every positive integer q ≤ min(m, n), I claim that the space A_{q} of m × n matrices with support in the union of the first q rows and the first q columns, such that the first q entries along the main diagonal are all equal, satisfies (1) and (2). To see (1), note that given any nonzero column vector b = (β_{1}, …, β_{m})^{T} , we can find a nonzero row vector c = (γ_{1}, …, γ_{q}, 0, …, 0) such that β_{1}γ_{1} = ⋯ = β_{q}γ_{q}. Indeed, if at least one of β_{1}, …, β_{q} is zero, we can choose the γ’s not all zero so that all products β_{i}γ_{i} are zero, while if all of β_{1}, …, β_{q} are nonzero, we can choose the γ’s so that all β_{i}γ_{i} equal 1. In either case, the m × n matrix b ⊗ c will be nonzero and lie in A_{q}. Similarly, given a nonzero row vector c, we can find a nonzero column vector b such that b ⊗ c ∈ A_{q}, proving (2).

It is not hard to see that for every column vector b ∈ k^{m} at most one of whose first q entries is zero, and which has at least one nonzero entry after the first q entries, the above construction gives (up to scalars) the only c such that b ⊗ c ∈ A_{q}. (The condition that at least one entry of b after the first q be nonzero guarantees that every c with b ⊗ c ∈ A_{q} must live in its first q entries, which we need to get uniqueness.) Likewise, if we are given c at most one of whose first q entries is zero, and having at least one later nonzero entry, there is up to scalars only one b such that b ⊗ c ∈ A_{q}. Hence, any subspace of A_{q} that satisfies (1) and (2) must contain all the elements b ⊗ c of the two sorts just described. Combining these observations, one can deduce that if at least one of m, n is > q, and k has > 2 elements, then A_{q} indeed has no proper subspaces satisfying (1) and (2). (The condition that k has > 2 elements is used to get every row or column as a sum of rows or columns satisfying appropriate conditions on which entries are nonzero. For k the 2-element field and q > 2, there do in fact exist proper subspaces of A_{q} satisfying (1) and (2).) In the remaining case, m = n = q, one finds that the same argument works without the proviso in the first sentence of this paragraph about “at least one nonzero entry after the first q entries”.

So if k has more than 2 elements, these constructions do give minimal examples. I don’t know whether, conversely, every minimal subspace of a tensor product B ⊗_{k} C satisfying (1) and (2) is, with respect to some bases of B and C, of one of these forms.

It is curious that the bound dim(A) ≥ dim(B) + dim(C) − 1 of Proposition 1 also appears in [2, Proposition 1.3, case k = 1], for subspaces A ⊆ B ⊗ C subject to a different condition involving rank-1 entities; namely, that A not be contained in the kernel of any tensor product f ⊗ g of linear functionals f ∈ A^{∗}, g ∈ B^{∗} [2, Proposition-Definition 1.1(2), case k = 1]. But I cannot see a relation between the two results. Indeed, the result in [2] is proved only over an algebraically closed field, and it is noted that it fails without that condition; while the last clause of [2, Proposition 1.3] implies that every minimal submodule A of the sort considered there has dimension exactly dim(B)+dim(C)−1, though for our condition, we have seen the opposite. Finally, most of the minimal examples noted above for our condition do not satisfy the condition of [2], precisely because their matrix representations involve 0’s in fixed locations. So if there is a relation between the two results, it must be a subtle one.

Turning back to Theorem 3, and restricting attention to commutative R, observe that if we wish to extend that result to a commutative not-necessarily-local Artinian ring R, then this will be a direct product of commutative local rings e_{1}R × ⋯ × e_{d}R, where the e_{i} are the minimal idempotents of R. By summing the inequalities (5) for these d rings, we get a similar inequality, with the final +1 replaced by +d. The dimensions in the inequalities we sum are with respect to different base fields e_{i}R/J(e_{i}R); the most natural way to refer to these dimensions is as the lengths of those modules. So in this situation, the analog of (5) is

(6)  length(M/J(R)M) + length(socle(M)) ≤ length(socle(R)) + d.

This points toward the form of the inequalities we will obtain in the next two sections, for not-necessarily-commutative Artinian R.

3. The general Artinian case – preliminary steps

The result we are now aiming for will again be an application of a statement about a bilinear map, but with semisimple left modules in the roles of B and C, and a bimodule in the role of A. In this case, I don’t see how to turn statements about bilinear maps into statements about subobjects of tensor products, so we shall develop directly the “ A × B → C ” result analogous to Corollary 2. (However, there will be an obvious symmetry between what we do with B and with C; which suggests that I am missing some way that these can be unified.)

Throughout this and the next section, we will therefore assume that

(7)

S and T are semisimple Artinian rings, _{S}B and _{T} C are left modules of finite lengths, _{T} A_{S} is a bimodule, and h : _{T} A_{S} ×_{S}B → _{T} C a balanced bilinear map,

such that

(8) Every nonzero a ∈ A induces a nonzero map of abelian groups h(a, −) : B → C.

In our final application, S and T will be the same, namely R/J(R); but keeping them distinct until then will make our manipulations clearer.

For a ∈ A, we shall think of h(a, −) : B → C as “the action of a ”, and thus speak of elements of A as annihilating certain elements of B, having certain elements of C in their images, etc.; thus, we will seldom mention h explicitly.

When we speak of the kernel or image of a family of maps, we shall mean the intersection of the kernels of those maps, respectively, the sum of their images. A “maximal submodule” of a module will mean a maximal proper submodule.

We now state the conditions corresponding to (1) and (2).

(9)

For every maximal submodule B_{0} ⊆ B, there exists a nonzero a ∈ A which annihilates B_{0}; equivalently, such that the kernel of a S is precisely B_{0}.

(10)

For every simple submodule C_{0} ⊆ C, there exists a nonzero a ∈ A which carries B into C_{0}; equivalently, such that the image of T a is precisely C_{0}.

The statements of equivalence follow by combining the fact that ker(a S) and im(T a) are submodules of _{S}B and _{T} C with the assumptions that B_{0} is maximal and C_{0} simple.

Let us now see what happens to conditions (9) and (10) when we write S and T as direct products of simple Artin rings, and decompose A, B and C accordingly. Say the decompositions of the identity elements of S and T into minimal central idempotents are 1_{S} = e_{1} + ⋯ + e_{m} and 1_{T} = f_{1} + ⋯ + f_{n}, so that S ≅ ∏_{i} e_{i}S and T ≅ ∏_{j} f_{j}T as rings. Then in the situation of (10), the simple submodule C_{0} is necessarily contained in some summand f_{j} C, so the a that we get must satisfy a = f_{j} a. Moreover, if we take some e_{i} such that a e_{i} ≠ 0, then the image of a e_{i} must also generate the simple submodule C_{0}; so replacing a by a e_{i} if necessary, we can assume a ∈ f_{j} A e_{i}.

Let us now fix some f_{j}. Then we can deduce from (10) and the above observations that every simple submodule of f_{j} C is generated by the image of an element of f_{j} A e_{i} for some i ≤ m. It would be nice if we could reverse the order of quantifications, and say that there exists some i ≤ m, such that every simple submodule of f_{j} C is generated by the image of an element of f_{j} A e_{i}; but that is too much to hope for. However, it turns out that, under some weak assumptions, we can show that there exists an i such that enough of the simple submodules of f_{j} C are images of elements of f_{j} A e_{i} to suit our purposes. The key concept is introduced in the definition below. In that definition, think of X as either B or e_{i} B, of Y as f_{j} C, of W as f_{j} A, or f_{j} A e_{i}, and of V as f_{j} T f_{j}. The adjective “left” in that definition, and “right” in the one that follows it, refer to whether we are thinking of fixing the left factor f_{j} in the product-symbol f_{j} A e_{i} (as in the above discussion), or the right factor e_{i}.

Definition 4. If W is a set of homomorphisms from an abelian group X to a left module Y over a simple Artin ring V, and N is a positive real number, then we shall call W left N-strong if for every family of ≤ N proper submodules of Y, there exists w ∈ W such that V w(X) is a simple submodule of Y not contained in any member of that family. We shall call W left strong if it is left N-strong for all positive real numbers N. We shall at times use “left ∞-strong” as a synonym for “left strong”.

(We have taken N, when finite, to be a real number rather than an integer so that some later statements can be formulated more conveniently; e.g., so that we can say “N/d-strong” rather than “⌊N/d⌋-strong”.)

We similarly define a variant of condition (9), namely

Definition 5. If W is a set of homomorphisms from a left module X over a simple Artin ring U to an abelian group Y, and N a positive real number, then we shall call W right N-strong if for every family of ≤ N nonzero submodules of X, there exists w ∈ W such that ker(w U) is a maximal submodule of X, and does not contain any member of that family. We shall call W right strong if it is right N-strong for all positive real numbers N. We shall at times use “right ∞-strong” as a synonym for “right strong”.

The observations following (9) and (10) will yield statements to the effect that for appropriate N, the sets ⋃_{i} f_{j} A e_{i} of maps B → f_{j} C are left N-strong, and that the sets ⋃_{j} f_{j} A e_{i} of maps e_{i} B → C are right N-strong. (We postpone the details for the moment.) The next lemma shows the virtue of the N-strong condition: it is inherited, in slightly weakened form, by at least one term of any such union.

Lemma 6. Suppose, in the context of Definition 4, that W is the union W_{1} ∪ ⋯ ∪ W_{d} of d subsets. Then if W is left N-strong, for N a positive real number or ∞, then at least one of the W_{i} is left N/d-strong. (So for N = ∞, this says that if W is left strong, so is one of the W_{i}.)

Likewise, if in the context of Definition 5, W = W_{1} ∪ ⋯ ∪ W_{d} is right N-strong, then at least one of the W_{i} is right N/d-strong.

Proof. Let us prove the statements for finite N in contrapositive form. In the case of the first assertion, if left N/d-strength fails for each of the W_{i}, then for each i we can find a family of ≤ N/d proper submodules of Y whose union contains all simple submodules generated by images of members of W_{i}. Taking the union over i of these sets of submodules, we get a set of ≤ N proper submodules of Y whose union contains all such simple submodules, showing that W is not left N-strong. The dual argument works for the second statement. The cases with N = ∞ follow from the cases for finite N. □

The next lemma gives the postponed argument which will allow us to obtain from conditions (9) and (10) statements about N-strength; i.e., which will show that “all simple submodules” as in (10) entails “a family of submodules not contained in the union of N proper submodules”, for appropriate N.

We have spoken above of the minimal central idempotents of a semisimple ring T; we shall now deal with minimal idempotents (where an idempotent e of a ring R is considered less than or equal to an idempotent f if eRe ⊆ fRf; equivalently, if e = ef = fe. By “minimal” we of course mean “minimal nonzero”.) Note that if f is a minimal central idempotent of a semisimple artinian ring T, then V = f T f is a simple Artinian ring, that is, a matrix ring Matr_{n,n}(D) over some division ring D [4, Theorem 3.5]; and that by taking any minimal idempotent f′ ∈ Matr_{n,n}(D), one can recover D up to isomorphism as f′ Matr_{n,n}(D) f′ = f′(f T f)f′ = f′T f′.

Lemma 7. Let V be a simple Artin ring, Y a left V -module, f′ a minimal idempotent of V, and N = card(f′ V f′).

Then there exists no finite family of proper submodules Y_{1}, …, Y_{n} of Y with n ≤ N such that every simple submodule of Y is contained in some Y_{i}.

Likewise, there exists no finite family of nonzero submodules Y_{1}, …, Y_{n} of Y with n ≤ N such that every maximal submodule contains some Y_{i}.

Proof. Note that both conclusions are statements about the lattice of submodules of Y. Now letting D = f′ V f′, a division ring, we know that V is Morita equivalent to D, so the lattice of submodules of Y is isomorphic to that of the D-vector-space f′Y. Hence in our proof, we may assume without loss of generality that V is a division ring D, and Y a D-vector space.

In this situation, the simple submodules are just the cyclic submodules, so to prove the first statement, we have to show that given proper subspaces Y_{1}, …, Y_{n} of Y with n ≤ N, there exists a y ∈ Y which is not contained in any of the Y_{i}. Suppose inductively that for some m < n we have found a y ∈ Y which does not lie in any of Y_{1}, …, Y_{m}. If y also does not lie in Y_{m+1}, we have our next inductive step. If, on the other hand, y ∈ Y_{m+1}, take any y′ ∉ Y_{m+1}, and consider the N elements y′ + αy, as α runs over D. Clearly, none of these lie in Y_{m+1}, and it is easy to check that at most one can lie in each of Y_{1}, …, Y_{m}. Since they constitute N ≥ n > m elements, at least one lies in none of these spaces, giving the inductive step.

To get the second statement, recall that maximal subspaces of Y are kernels of elements of the dual space Y^{∗} = Hom(Y,_{D}D), a right D-vector-space. Hence we can apply to that space the left-right dual of the first statement, and get the desired result. □

The next corollary applies the two preceding results to the modules and map of (7). Note that the statement refers to both minimal idempotents and minimal central idempotents.

Corollary 8. Suppose h : _{T} A_{S} ×_{S}B → _{T} C is as in (7), and satisfies (8), (9), and (10).

Let N_{T} be the minimum of the cardinalities of the division rings f T f as f ranges over the minimal idempotents of T, if that minimum is finite, or the symbol ∞ if all those division rings are infinite. Let d_{T} be the maximum, as f ranges over the minimal central idempotents of T, of the number of minimal central idempotents e of S such that fA e ≠ 0. Then for each minimal idempotent f of T, there is at least one minimal idempotent e of S such that fA e is left N_{T} /d_{T} -strong as a set of maps e B → f C.

Likewise, let N_{S} be the minimum of the cardinalities of the division rings eSe as e ranges over the minimal idempotents of S if this is finite, or the symbol ∞ if that minimum is infinite, and d_{S} the maximum, as e ranges over the minimal central idempotents of S, of the number of minimal central idempotents f of T such that fA e ≠ 0. Then for each minimal idempotent e of S, there is at least one minimal idempotent f of T such that fA e is right N_{S}/d_{S}-strong as a set of maps e B → f C.

Proof. In the situation of the first assertion, we see from condition (10) that for each minimal central idempotent f of T, the set f A of morphisms f a : B → f C (a ∈ A) has the property that for every minimal f T f-submodule C_{0} ⊆ f C, there is at least one nonzero f a ∈ f A which takes B into C_{0}. Now the minimal central idempotents e ∈ S sum to 1, so there is at least one such e ∈ S such that f a e ≠ 0. This map clearly still carries B into C_{0}; so we conclude that for every minimal C_{0} ⊆ f C, some nonzero element of ⋃_{e}(fA e − {0}) has image in C_{0}, where the union is over the minimal central idempotents e of S. By Lemma 7, this shows that if we write D for the division ring (unique up to isomorphism) given by f′T f′ where f′ is any minimal idempotent of f T f, then ⋃_{e}(fA e − {0}) is left card(D)-strong; hence left N_{T} -strong.

Now by definition of d_{T} , there are at most d_{T} minimal central idempotents e such that fA e ≠ 0; so ⋃_{e}(fA e − {0}) involves at most d_{T} nonempty sets fA e − {0}. Hence by Lemma 6, at least one of these sets is left N_{T} /d_{T} -strong, as claimed.

The second assertion is proved in the analogous fashion. □

The above corollary will enable us to prove our desired generalizations of (3) and (5) unless one or more of the division rings eSe and f T f are finite fields of small cardinality. (If such a field occurs, N_{T} /d_{T} or N_{S}/d_{S} may be too small for our arguments to work.) It is curious that a similar condition in my original proof of (3) and (5) was eliminated by de Seguins Pazzis’s argument; but we shall see by example, in §5, that the corresponding condition in the present situation cannot be dropped.

While these cardinality conditions are not very restrictive, and are needed for the result to hold, we come now to an embarrassingly restrictive condition, needed for the proofs of the results of the next section, though I have no example showing that those results fail without it. The condition is awkward to state in maximum generality. A fairly natural special case is the hypothesis that in (7),

(11)

The semisimple Artinian rings S and T are finite-dimensional algebras over a common algebraically closed field k, and the induced actions of k on the two sides of the bimodule _{T} A_{S} are the same.

Actually, we need the assumption that k is algebraically closed only to make the simple factors of S and T full matrix algebras over k; so we can instead put that assumption on the table, as the condition

(12)

For some field k, each of the semisimple Artinian rings S and T is a direct product of full matrix algebras over k, and the induced actions of k on the two sides of the bimodule _{T} A_{S} are the same.

A condition that is still more general (as we shall show in the next lemma), and will suffice for our purposes, is

(13)

If a ∈ A is a nonzero element whose image a B is contained in a simple submodule of C, then there exists nonzero a′ ∈ T aS whose kernel contains a maximal submodule of B; and likewise if a ∈ A is a nonzero element whose kernel contains a maximal submodule of B, then there exists nonzero a′ ∈ T aS whose image is contained in a simple submodule of C.

In fact, we can make do with the following still weaker (though still wordier) condition.

(14)

For each minimal central idempotent e ∈ S and minimal central idempotent f ∈ T such that f A e ≠ {0}, it is either true that for every a ∈ fA e such that the induced map e B → f C has image in a simple submodule of f C, some nonzero a′ ∈ T aS has kernel containing a maximal submodule of B, or that for every a ∈ fA e such that the induced map e B → f C has kernel containing a maximal submodule of B, some nonzero a′ ∈ T aS has image in a simple submodule of f C. (But which of these is true may vary with the pair (f, e).)

Let us note the implications among these conditions.

Lemma 9. For h : _{T} A_{S} ×_{S}B → _{T} C a bilinear map as in (7) which satisfies (8), one has the implications (11) ⟹ (12) ⟹ (13) ⟹ (14).

Proof. (11) ⟹ (12) follows from the standard description of the structures of semisimple Artin rings, and (13) ⟹ (14) is clear. Let us prove (12) ⟹ (13).

Suppose as in (13) that a ≠ 0, and a B is contained in a simple submodule C_{0} ⊆ C. Since the identity elements of S and T are sums of minimal central idempotents, we can find such idempotents e ∈ S and f ∈ T such that f a e ≠ 0; and we will have f a e B ⊆ f a B ⊆ f C_{0} ⊆ C_{0}. Hence, replacing a by some a′ = f a e if necessary, we may assume without loss of generality that a ∈ fA e for such a pair of idempotents.

Now by assumption, eSe and f T f have the forms Matr_{m,m}(k) and Matr_{n,n}(k) for some positive integers m and n. Identifying them with these matrix rings, it is easy to verify that there exist finite-dimensional k-vector-spaces B′, C′ such that we can identify eB and fC with the spaces B^{′m} and C^{′n} of column vectors of elements of B′ and C′, made into modules over eSe = Matr_{m,m}(k) and fT f = Matr_{n,n}(k) in the natural way; that fAe can then be identified with Matr_{n,m}(A′) where A′ is a k-vector-space of k-linear maps B′ → C′, and finally, that the simple submodule C_{0} ⊆ f C will have the form C_{0}^{′n} for some 1-dimensional subspace C′_{0} ⊆ C′. (Explicitly, letting e′ and f′ denote minimal idempotents of S and T, say those given by the matrix units e_{11} of their matrix representations, we can take B′ = e′B, C′ = f′C, C′_{0} = f′C_{0}, and A′ = f′A e′.)

Thus, our element a ∈ fA e will be an n × m matrix of linear maps e B′ → f C′, each having range in C′_{0}. We can now choose ā ∈ T aS − {0} to be nonzero and have all components in some 1-dimensional subspace k a′ ⊆ A′. (E.g., we can let ā be a product e_{1,i} a e_{j,1} such that the (i, j) component of a is nonzero.) Since a′ : B′ → C′ has range in the 1-dimensional subspace C′_{0} of C′, it is a rank-1 k-linear map, hence has kernel B′_{0} ⊆ B′ of codimension 1. Hence ā, having all components in k a′, will have kernel containing the maximal proper submodule (B′_{0})^{m} ⊆ B, as required.

The second assertion of (13) is proved similarly. □

4. Minimal faithful modules over left Artin rings

Recall that if G is a finite graph, its Euler characteristic χ(G) is the number of vertices of G minus the number of edges, an integer which may be positive, negative or zero. Recall also that a bipartite graph is a graph whose vertex-set is given as the disjoint union of two specified sets, such that every edge connects a member of one set with a member of the other. We shall call those sets (nonstandardly) the left and right vertex-sets of G.

The hard work of this section comes right at the beginning: proving the following noncommutative analog of Proposition 1, or more precisely, of Corollary 2.

Proposition 10. Suppose h : _{T} A_{S} ×_{S}B → _{T} C is a bilinear map as in (7), which satisfies (8), (9), (10) and (14). For notational convenience we shall assume S and T disjoint.

Let N_{S}, d_{S}, N_{T} and d_{T} be defined as in Corollary 8. Further, let l_{S} denote the maximum of the values length_{eSe}(e B) as e ranges over the minimal central idempotents e ∈ S, and l_{T} the maximum of length_{f T f}(f C) as f ranges over the minimal central idempotents f ∈ T; and assume that

(15) N_{T} ≥ d_{T} l_{S}, and N_{S} ≥ d_{S} l_{T} .

Finally, let G be the bipartite graph whose right vertex-set is the set of minimal central idempotents e ∈ S satisfying e B ≠ 0 (equivalently, A e ≠ 0), whose left vertex-set is the set of minimal central idempotents f ∈ T satisfying f C ≠ 0 (equivalently, f A ≠ 0), and such that two such vertices e, f are connected by an edge (f, e) if and only if fA e ≠ {0}.

Then

(16) length(_{S}B) + length(_{T} C) ≤ length(_{T} A_{S}) + χ(G).

Proof. The parenthetical equivalences in the definition of G follow from (8), (9) and (10). Combining Corollary 8 with our hypothesis (15), we find that

(17)

For each minimal idempotent f of T, there is at least one minimal idempotent e of S such that fA e is left l_{S}-strong as a set of maps e B → f C; and for each minimal idempotent e of S, there is at least one minimal idempotent f of T such that fA e is right l_{T} -strong as a set of maps e B → f C.

We shall now perform a series of reductions and decompositions on our system _{T} A_{S} ×_{S}B → _{T} C, verifying at each stage that if the inequality corresponding to (16) holds for our simplified system(s), then it also holds for the original system; and, finally, we shall establish that inequality for the very simple sorts of system we end up with.

In preparation, let us harness (17) by choosing, arbitrarily, for each minimal central idempotent f ∈ T, one minimal central idempotent e ∈ S such that fA e is left l_{S}-strong, and call (f, e) the left-marked edge of the graph G associated with the vertex f; and similarly, for each minimal central idempotent e ∈ S, let us choose a minimal central idempotent f ∈ T such that fA e is right l_{T} -strong, and call (f, e) the right-marked edge of G associated with the vertex e. Some edges may be both right- and left-marked (for their respective right and left vertices).

We begin our reductions by considering any edge (f, e) ∈ G which is neither right- nor left-marked, and seeing what happens if we drop the summand fA e from A; i.e., replace A with (1 − f)A + A(1 − e); and thus drop the edge (f, e) from G, leaving the rest of our system unchanged. Because (f, e) is neither right nor left marked, condition (17) has not been lost. (The constant d_{S} and/or d_{T} may have decreased by 1, but we don’t have to think about this, because our use of these constants was only to obtain (17), which has been preserved.) The removal of fA e has no effect on the left-hand side of (16), while on the right-hand side, it decreases length(_{T} A_{S}) by length(_{T} (fA e)_{S}) ≥ 1, and increases χ(G) by 1. Hence if the new system satisfies (16), then the original system, whose right-hand side is ≥ that of the new system, also did.

Hence by induction, the task of proving (16) is reduced to the case where every edge of G is left and/or right marked.

Suppose, next, that there is some left vertex f of G such that the only edge adjacent to f is its associated left-marked edge, say (f, e), and such that this is not also right-marked, and consider what happens if we remove both the vertex f and the edge (f, e); i.e., replace C by (1 − f)C, and A by (1 − f)A.

Clearly, the remaining vertices and edges continue to witness condition (17). Our new system also has the same Euler characteristic as the old one, since just one vertex and one edge have been removed from the graph.

To see how (16) is affected, let d = length(_{T} f C). I claim that we can find elements a_{1}, …, a_{d} ∈ fA e such that the submodules T a_{i} B ⊆ f C are simple and their sum is direct. Indeed, assuming we have constructed a_{1}, …, a_{j} with j < d, the submodule ⨁_{i≤j} T a_{i} B ⊆ f C will have length ≤ j < d = length(_{T} f C), so the fact that fA e is left 1-strong (a weak consequence of the fact that (f, e) is left marked, hence that fA e is left l_{S}-strong) allows us to find a_{j+1} with T a_{j+1} B simple and not contained in that proper submodule; and since it is simple, its sum with that submodule is direct. Once we have a_{1}, …, a_{d}, we see that the subbimodules ∑_{i≤j} T a_{i} S ⊆ A (j ≤ d) form a chain of length d. Hence length(_{T} (fA e)_{S}) ≥ d = length(_{T} f C), from which we see that replacing A with (1 − f)A decreases the right-hand side of (16) by at least as much as replacing C with (1 − f)C decreases the left-hand side. So again, if the new system satisfies (16), so did the old one.

Similarly, if G has some right vertex e such that the only edge adjacent to e is its associated rightmarked edge (f, e), and this edge is not also left-marked, we find that we can remove fA e from A, and e B from B, and that if the resulting system satisfies (16), so does our original system. The proof is the same, except that where above we used an increasing family of submodules ⨁_{i≤j} T a_{i} B ⊆ f C, we now use a decreasing family of submodules ⋂_{i≤j} ker(a_{i} S) ⊆ e B.

Repeating these two kinds of reductions until no more instances are possible, we are left with a system in which every vertex not only hosts its own marked edge, but also hosts the marked edge of at least one other vertex (which may or may not be the same as its own marked edge). By counting vertices, we immediately see that in the preceding sentence, “at least one” can be replaced by “exactly one”. From this, it is easy to see that each connected component of G is now either a loop of even length > 2, or a graph having just two vertices, and a single edge which is marked for both of these.

It follows that our system _{T} A_{S} ×_{S}B → _{T} C decomposes into a direct sum of subsystems corresponding to those connected components, and that (16) will be the sum of the corresponding inequalities for those components. Hence, it will suffice to prove (16) in the two cases where G is a loop, and where G has just a single edge. The l_{S} and l_{T} for each such system are ≤ the l_{S} and l_{T} for our original system, so (17) will hold for these systems, because it held for the original system.

If G is a loop, then each edge is marked for only one vertex, and the proof is quick: The Euler characteristic χ(G) is zero, while for each vertex, the bimodule corresponding to its marked edge has at least the length of the module corresponding to that vertex, by the same “ ∑_{i≤j} T a_{i} S ” arguments used in the preceding reduction. Summing these inequalities over the vertices, we have (16).

We are left with the case where G has just one right vertex, e, one left vertex, f, and the single edge (f, e). Thus, B = eB and C = fC. Let

(18) m = length(_{S}B),  n = length(_{T} C).

Note that the fact that (f, e) is both left- and right-marked tells us that A is both left l_{S}-strong and right l_{T} -strong.

It is now that we will use our hypothesis (14). Let us begin by assuming the second of the two alternatives it offers, which in this situation says that whenever an a ∈ A has kernel containing a maximal submodule of B, then some nonzero element of T aS has image in a simple submodule of C. Under this assumption, we begin by constructing, for m as in (18), elements a_{1}, …, a_{m} ∈ A such that

(19)

T a_{1} S, …, T a_{m} S have for kernels maximal submodules of B, none of which contains the intersection of the kernels of the others, and each T a_{i} S has for image a simple submodule of C.

To see that we can do this, suppose inductively that we have constructed i < m elements a_{1}, …, a_{i} with these properties. Since i < m = length(_{S}B), the intersection of the kernels of T a_{1} S, …, T a_{i} S is a nonzero submodule of B, hence since A, being right l_{T} -strong, is in particular right 1-strong, we can find a ∈ A such that the kernel of a S is a maximal submodule B_{0} ⊆ B and does not contain that intersection. Now any a′ ∈ T a_{i} S − {0} will still annihilate B_{0}; hence, if it is nonzero, T a′ S must have exactly that annihilator. By our assumption from (14), we can find such a nonzero a′ which has image in a simple submodule of C. Taking for a_{i+1} this element, we have our desired inductive step; for the intersection of the kernels of T a_{1} S, …, T a_{i+1}S has co-length i + 1 in B, hence none can contain the intersection of the kernels of the others. Thus, we get (19).

I claim next that we can find n − 1 more elements, a′_{1}, …, a′_{n−1} ∈ A, where each a′_{j} satisfies

(20)

T a′_{j} B is a simple submodule of C, and for each i = 1, …, m, we have

T a′_{j} B ⊈ T a_{i} B + T a′_{1} B + ⋯ + T a′_{j−1} B.

To see this, suppose inductively that a′_{1}, …, a′_{j−1} have been chosen. For each i ≤ m, both T a_{i} B and each of T a′_{1} B, …, T a′_{j−1} B are simple submodules of C, and there are 1 + (j − 1) = j < n of these, so their sum is a proper submodule. As i ranges from 1 to m we get m = l_{S} such proper submodules; so as A is left l_{S}-strong, our a′_{j} can be chosen to satisfy (20).

Let us now show that as we add up successively the subbimodules T a_{1} S + ⋯ + T a_{m} S + T a′_{1} S + ⋯ + T a′_{n−1} S of A, we get a chain of length m + n − 1. The first m steps are distinct by comparison of their kernels in B, in view of (19). If we had equality somewhere in the next n − 1 steps, this would mean that for some j < n we would have

(21) T a′_{j} S ⊆ T a_{1} S + ⋯ + T a_{m} S + T a′_{1} S + ⋯ + T a′_{j−1} S.

To get a contradiction, let us, for i = 1, …, m, write B_{i} for the intersection of the kernels of all the T a_{j} S other than T a_{i} S. By (19), each B_{i} has co-length m − 1, hence is a simple submodule of B, and no B_{i} is contained in the sum of the others; so ∑_{i} B_{i} = B. Hence, some B_{i} is not in the kernel of T a′_{j}S; let us choose such a B_{i} and apply (21) to it. We get, on the left, T a′_{j}B_{i}, which by our choice of i is nonzero. Since it is contained in T a′_{j}B, which by (20) is simple, it must be equal thereto. On the right, by definition of B_{i}, we loose all of the first m terms other than the i-th. Replacing the remaining occurrences of B_{i} on the right by the larger module B, we get

(22) T a′_{j}B ⊆ T a_{i} B + T a′_{1} B + … + T a′_{j−1} B.

But this is one of the relations we chose a′_{j} to avoid in (20). This contradiction proves that our chain of submodules of A is strictly increasing, hence that A has length at least m + n − 1 = length(_{S}B) + length(_{T} C) − χ(G), as required.

If we are in the other case of (14), we operate dually, and first obtain n elements a_{1}, …, a_{n} of A which determine independent simple submodules of C, and each of which has kernel containing a maximal submodule of B, then choose m − 1 more elements a′_{1}, …, a′_{m−1}, such that the submodule of B which each determines is maximal, and does not contain the intersection of the submodules determined by the proceeding members of that list, intersected with the submodule determined by any one of the a_{i}. □

We can now get our main result. Since I don’t see how to turn (13) or (14) into a condition on the ring R, I will only give the hypothesis corresponding to (12).

Condition (15) would put a finite lower bound on the size of k; but since this bound would involve both the structure of R and that of M, and we would like our restrictions on M to be stated in terms of the structure of R, I will achieve this simply by requiring k to be infinite.

The two rings S and T of the preceding development are the same ring R/J(R) in the application below, so we shall distinguish the corresponding sorts of vertices of our graph with subscripts “left” and “right”. We shall regard socle(R) (the 2-sided socle of R, whose definition we recalled in the paragraph before Theorem 3) as a bimodule over R/J(R); as such it will play the role of the A of Proposition 10.

Theorem 11. Suppose k is an infinite field and R an overring of k, such that R is finite-dimensional as a left k-vector-space, k has central image in R/J(R) and centralizes socle(R), and R/J(R) is a direct product of full matrix rings over k (this last condition being automatic if k is algebraically closed).

Let G be the bipartite graph whose left vertex-set consists of symbols f_{left} for all minimal central idempotents f ∈ R/J(R) such that f socle(R) ≠ {0}, whose right vertex-set consists of symbols e_{right} for all minimal central idempotents e ∈ R/J(R) such that socle(R) e ≠ {0}, and whose edge-set is {(f_{left}, e_{right}) | f socle(R) e ≠ {0} }.

Then for any faithful left R-module M such that no proper submodule or proper homomorphic image of M is faithful, we have

(23) length(M/J(R)M) + length(socle(M)) ≤ length(socle(R)) + χ(G),

where the two lengths on the left are as left R-modules, while the length on the right is as an (R, R)-bimodule.

Sketch of proof. Under the action of R on M, elements of socle(R) annihilate J(R)M and have image in socle(M); so that action induces a balanced bilinear map of R-modules and bimodules, socle(R) × M/J(R)M → socle(M). Since all four R-module structures involved (the three left module structures, and the right module structure of socle(R)) are annihilated by J(R), that induced operation is a map of the form (7), with R/J(R) in the roles of both S and T. Since M is faithful, this map satisfies (8), while as in the proof of Theorem 3, the minimality assumptions on M give (9) and (10). Since (12) implies (14), we can apply the preceding proposition, getting (23). □

5. Examples, remarks, and questions

5.1. Counterexamples over small fields. To show that Proposition 10 can fail if condition (15) (the requirement that our division rings not be too small) is dropped, let k be the field of 2 elements, let A = B = k × k × k and C = k × k (as abelian groups for the moment), and define h : A × B → C by h((α_{1}, α_{2}, α_{3}), (β_{1}, β_{2}, β_{3})) = α_{1}β_{1}(1, 0) + α_{2}β_{2}(0, 1) + α_{3}β_{3}(1, 1). Letting S = k × k × k and T = k, and defining the module structures of _{T} A_{S}, _{S}B and _{T} C in the obvious ways, we see that h is indeed a balanced bilinear map, i.e., satisfies (7).

This map also clearly satisfies (12) and (8), and it is not hard to verify (9) and (10) as well. However, (15) fails, since N_{T} = 2, d_{T} = 3, l_{S} = 1. And in fact, the conclusion (16) fails: the left-hand side of that inequality is 3 + 2, while the right-hand side is 3 + 1. If we examine the steps of our proof of Proposition 10 in this case, we see that conditions (9) and (10) make A, regarded as a family of maps B → C, left 2-strong, but of its three components A e_{i} : e_{i} B → C, none is 1-strong. We can, as in the proof of that proposition, snip two leaves off G without changing the numerical relationship between the two sides of (16); but the remaining system, say A e_{1} : e_{1} B → C, does not satisfy (16).

More generally, over any finite field k, say of q elements, one can get a similar example by taking S = A = B = k^{q+1}, T = k, and C = k^{2}, and letting the q + 1 components A e_{i} : B → C have for images the q + 1 one-dimensional subspaces of C. (The reader who has worked through the above q = 2 case should not find it hard to supply the details for this generalization.) Still more generally, if we take C to be d-dimensional (d ≥ 2), we can make S = A = B = k^{(q^{d}−1)/(q−1)}, and let the natural basis of A act by maps having for images the (q^{d} − 1)/(q − 1) one-dimensional subspaces of C.

We can adapt these constructions to get examples showing that for small k Theorem 11 likewise fails; but since in that case S and T must be the same, a bit of adjustment is needed. We can keep S as in those constructions, but let T = S, giving it actions on C and A under which all but one of its minimal idempotents annihilate those objects.

Let me describe in concrete terms an R and M based, in this way, on our initial example where k is the field of 2 elements. Our R will be the ring of all 4×4 matrices over this k with support in the union of the first row and the main diagonal, and whose (1, 1) and (2, 2) entries are equal. To obtain M, we start with the left R-submodule of Matr_{4,3}(k) spanned as a k-vector-space by {e_{11}, e_{12}, e_{13}, e_{21}, e_{32}, e_{43}}, and divide out by the subspace spanned by e_{11} +e_{12} +e_{13}. We find that J(R)M = socle(M) = the 2-dimensional space spanned by e_{11} and e_{12}; that M is a faithful R-module such that no proper submodule or homomorphic image of M is faithful, but that M does not satisfy (23), which for this case would say 3 + 2 ≤ 3 + 1.

5.2. An example not satisfying (14). I will give here an example of a system (7) arising as the map socle(R) × M/J(M) → socle(M) for a faithful module M over an Artinian local ring R, which does not satisfy (14), or our conclusion (16). In fact, it does not satisfy (9) or (10) either; but it may give some insight into how things can differ from the situation analyzed in the preceding sections.

Let K be the field ℚ(2^{1/3}), let F = K(ω), where ω is a primitive cube root of unity, and let σ be the automorphism of F of order 3 which fixes ω, and takes 2^{1/3} to ω 2^{1/3}. Let tr = tr_{F/K} : F → K be the trace operation. (Thus, tr is K-linear, but σ is not.)

Let M be the ℚ-vector-space F^{2}, and let us define the right shift operation M → M,

(24) s : (a, b) ↦ (0, a).

We shall understand tr, σ and each element of F to act on M componentwise. (So if a ∈ F, the symbol a will also represent the operation of componentwise multiplication of elements of M by a, which does not in general commute with either tr or σ.)

We now define two operations on M,

(25)

x = tr σ s : (a, b) ↦ (0, tr(σ(a))),
y = σ tr s : (a, b) ↦ (0, σ(tr(a))).

From the fact that [F : K] = 2, it is easy to see that K σ(K) = F = K σ^{−1}(K). Combining this with the fact that tr and s commute with the action of elements of K, we see that

(26) K x K = K tr σ s K = tr K σ s K = tr σ (σ^{−1}(K)) s K = tr σ s (σ^{−1}(K) K) = x F, K y K = K σ tr s K = K σ tr K s = K σ K tr s = (K σ(K)) σ tr s = F y.

These calculation show in particular that the bimodule operations of the (K, K)-bimodules spanned by each of x and y contain the operations of a 1-dimensional F -module; so each of these bimodules is simple.

Now let R be the ring of ℚ-vector-space endomorphisms of M generated by the actions of the elements of K, together with the two endomorphisms x and y. We see that

(27) R = K + K x K + K y K = K + x F + F y, and J(R) = socle(R) = x F + F y.

The subideals of socle(R) generated by x and by y are isomorphic to one another as bimodules over R/J(R) ≅ K, namely, each is isomorphic to the (K, K)-bimodule F σ = σ F. But as systems of maps M/J(R)M → socle(M) they behave differently: xF has range in (0, K tr(F)) = (0, K) ≠ (0, F), but has trivial kernel; F y, dually, has all of (0, F) as range, but, identifying M/J(R)M with F, its kernel is the nontrivial K-subspace ker(tr).

5.3. An approach that probably doesn’t go anywhere. Theorem 11 does not cover the case where R is a general finite-dimensional algebra R over an infinite field k, since in that situation, R/J(R) need not be a direct product of matrix rings. But it is natural to ask: Can’t we start with such an R and an R-module M, extend scalars to the algebraic closure of k, and apply Theorem 11 to the resulting structures?

The trouble is that the relevant properties of R and M may not carry over under this change of base field. I do not know whether we can expect the minimality conditions on M to carry over; but the set of minimal central idempotents of R can certainly grow under such an extension. So I do not see how anything can be achieved in this way.

5.4. Some ways our results can be strengthened. One step in our proof of Proposition 10 was noticeably wasteful. When we dropped all edges that were neither left nor right marked, we counted each of them as contributing “at least 1” to length(_{T} A_{S}). But depending on what we know about the structure of A, we may be able to raise this estimate. A difficulty is that knowing the structure of A doesn’t tell us which edges will be dropped, nor even exactly how many: the answers may depend on B and C.

However, we do know that dropping all unmarked edges must result in a graph in which there are at least as many vertices as edges (possibly more, if some edges are marked for both their vertices); so if the graph G determined by A has more edges than vertices, i.e., has χ(G) < 0, then at least −χ(G) edges (f, e) must be dropped. Thus, if we have in front of us a list of the lengths of all the nonzero bimodules fA e (with repetitions shown), then we can say that when we delete −χ(G) such bimodules, the sum of their lengths must be at least the sum of the −χ(G) smallest elements on that list. Deleting −χ(G) edges will leave us a system whose graph has Euler characteristic 0, to which we can apply Proposition 10 as it stands. Consequently, we have

Corollary 12. In the context of Proposition 10, let the family (set with multiplicity) of lengths of nonzero subbimodules fA e of A, listed in ascending order, be d_{1} ≤ d_{2} ≤ …. Then if χ(G) < 0, the term +χ(G) in (16) can be improved (decreased) to

(28) −d_{1} − d_{2} − ⋯ − d_{−χ(G)}.

The same applies, mutatis mutandis, to the inequality (23) of Theorem 11. □

One can do still a bit better, using the fact that we don’t delete as unmarked all the edges adjacent to any vertex. Consequently, if the lengths of the bimodules corresponding to edges adjacent to a certain vertex all lie among the first −χ(G) terms of the above sequence d_{1} ≤ d_{2} ≤ …, then in (28) we can skip the largest of these lengths, and instead throw in d_{−χ(G)+1}, which may be larger, at the end of the list; and iterate this process for other vertices.

One can also strengthen Theorem 11 by weakening the assumption that R contains k, to say that k is the residue field of a local ring contained in R, which again becomes central in R/J(R) and centralizes socle(R).

I have not put these observations into my formal statement of Theorem 11, because I feel that the more urgent task is to see whether one can generalize Proposition 10 to avoid or weaken the awkward restriction (14); and that if one can, some of these generalizations might be embraced by broader, more easily stated results.

5.5. Sketch of a more elaborate version of the N-strong condition. The reader may have noticed that in the proof of Proposition 10, the final case, where G has just one edge, both right and left marked, is roughly the situation of Proposition 1 (as reformulated in Corollary 2), but the proof is different from that of the earlier proposition. The reason is that the information we have available at that point in the proof of Proposition 10, that A is left l_{S}-strong and right l_{T} -strong, while useful in finding elements of A whose images lie in simple submodules of C outside of given submodules, and elements whose kernels have the dual property, does not provide a way of finding such elements whose images lie together in some proper submodule (and dually for kernels). But that was what we needed in Case 1 of the proof of Proposition 1, when we chose b_{1}, …, b_{m−1} ∈ ker(f).

However, I think the concept of an N-strong map can be modified to make it compatible with the method of proof of Proposition 1. Let me sketch how.

Given W, X and Y as in Definition 4, and adding the assumption that Y has finite length, our modified definition will involve a concept of W being left N-strong relative to a nonzero submodule Y′ ⊆ Y. That condition will be defined by recursion on the length of Y′. If Y′ has length 1, then W will be called left N-strong relative to Y′ if and only if W has a nonzero element whose image is contained in Y′. If the concept has been defined relative to submodules of some length r ≥ 1, then W will be said to be left N-strong relative to a submodule Y′ of length r + 1 if and only if for every submodule Y′′ ⊆ Y′ of length r − 1, there exist > N submodules of length r between Y′′ and Y′ relative to which W is left N-strong. We will simply say W is left N-strong (in our new sense) if it is left N-strong relative to its codomain Y.

It is now easy to verify that if, for every simple submodule of Y_{0} ⊆ Y, there is a nonzero element of W with image in Y_{0}, and if the division ring over which V is a full matrix ring has cardinality ≥ N, then W is left N-strong under our new definition. Moreover, an easy induction shows that if a union W = W_{1}∪⋯∪W_{d} is left N-strong, then at least one of the W_{i} is left N/d-strong.

The definition of right N-strong would be modified analogously.

In the proof of Proposition 1, the hypotheses (1) and (2) could then be weakened to say that (up to the change of notation appropriate to a subspace A ⊆ B ⊗_{k} C rather than a map A × B → C), A is both left and right 1-strong in our modified sense. I suspect that the same method could be adapted to the last part of the proof of Proposition 10, and would in fact allow us to weaken the hypothesis (15) by dropping the factors l_{S} and l_{T} .

I haven’t worked out the details, because they do not get at the serious restrictions in our results. In particular, the suggested change in the end of the proof of Proposition 10 would not get rid of the need for condition (14).

Let us now look at what we wish we could do.

5.6. Some questions. Here is an innocent-sounding generalization of our first main result that we might ask for.

Question 13. Does Theorem 3 remain true if the assumption that socle(R) is central in R is deleted?

To prove such a result, we would want a version of Proposition 1 involving a division ring rather than a field. As in the development of our results on non-local rings, I don’t see a convenient way of “symmetrizing” the general statement we need, so in the next question, I will ask, not for the analog of that proposition, but of its corollary. Moreover, although the result needed just to get a positive answer to Question 13 would have for B and C vector spaces over the same division ring, I expect it would be no more difficult to prove such a result without that restriction; so let us pose the question as follows.

Question 14. Let S and T be division rings, _{S}B and _{T} C be nonzero finite-dimensional vector spaces, and _{T} A_{S} be a subbimodule of the (T, S)-bimodule of all additive group homomorphisms _{S}B → _{T} C. Suppose moreover that

(29) For every proper subspace B′ ⊊ B, there is at least one a ∈ A whose restriction to B′ is zero.

and

(30)

For every proper homomorphic image C/C_{0} of C, there is at least one a ∈ A whose composite with the factor map C → C/C_{0} is zero.

Then must it be true that

(31) length(_{T} A_{S}) ≥ dim_{S}(B) + dim_{T} (C) − 1

(where length(_{T} A_{S}) denotes the length of A as a bimodule)?

I have posed this question in a relatively easy-to-state form, but my hope is that if a positive answer can be proved, then one will be able push the proof further, and get the same result with the hypotheses (29) and (30) weakened to say that A is left and right N-strong for appropriate N, in the modified sense sketched in §5.5 above (or something like it), and that this could be used to prove generalizations of Proposition 10 and Theorem 11.

Incidentally, it would be harmless to throw into our hoped-for variant of Proposition 10 the added assumption that A is semisimple as a left module, a right module, and a bimodule, since those conditions are true of the socle of a left Artin ring, the situation to which we apply that result in Theorem 11.

5.7. A waffle about wording. I would have preferred a more suggestive term for what I have called left and right N-strong families of maps. I was tempted to replace “strong” with “ubiquitous”; but this might suggest too much – a property like (9) and (10), rather than the weaker property actually defined. Another thought was “prevalent”; but this seemed a bit vague.

6. Getting minimal faithful modules from module decompositions

This section is essentially independent of the rest of this note, though the final assertion proved will complement the results proved above. Namely, Proposition 17(iii) below, though it has a weaker conclusion than Theorems 3 and 11, is applicable when the hypotheses of those theorems are not satisfied. Aside from that final result, the focus will be on modules with only one of the two minimality properties considered in preceding sections. (The one other exception to independence from the rest of this note is that at one point below, we will refer to an example from the preceding section.)

The two preliminary lemmas below may be of interest in their own right. The final statement of each describes how, under certain conditions, a module can be decomposed into “small pieces”: in the first, as a sum of submodules N such that N/J(R)N is simple; in the second, as a subdirect product of modules N with socle(N) simple.

Lemma 15. Let R be a left Artinian ring, and M a left R-module (not necessarily Artinian). Then

(i) If L is a simple submodule of M/J(R)M, then M has a submodule N such that the inclusion N ⊆ M induces an isomorphism N/J(R)N ≅ L ⊆ M/J(R)M.

Hence

(ii) Given a decomposition of M/J(R)M as a sum of simple modules ∑_{i∈I} L_{i}, one can write M as the sum of a family of submodules N_{i} (i ∈ I), such that for each i, N_{i}/J(R)N_{i} ≅ L_{i}, and L_{i} is the image of N_{i} in M/J(R)M.

Proof. In the situation of (i), let x be any element of M whose image in M/J(R)M is a nonzero member of L. Thus, the image of R x in M/J(R)M is L. Since R is left Artinian, so is R x, hence we can find a submodule N ⊆ R x minimal for having L as its image in M/J(R)M. Now since N/J(R)N is semisimple, it has a submodule L′ which maps isomorphically to L in M/J(R)M. If L′ were a proper submodule of N/J(R)N, then its inverse image in N would be a proper submodule of N which still mapped surjectively to L, contradicting the minimality of N. Hence N/J(R)N = L′ ≅ L, completing the proof of (i).

In the situation of (ii), choose for each L_{i} a submodule N_{i} ⊆ M as in (i). Then we see that M = J(R)M + ∑_{i} N_{i}, hence since R is Artinian, M = ∑_{i} N_{i} [4, Theorem 23.16 (1)⟹(2′)], as required. □

The next result is of a dual sort, but the arguments can be carried out in a much more general context, so that the result we are aiming for (the final sentence) looks like an afterthought.

Lemma 16. Let R be a ring and M a left R-module. Then

(i) If L is any submodule of M, then M has a homomorphic image M/N such that the composite map L ↪ M → M/N is an embedding, and the embedded image of L is essential in M/N (i.e., has nonzero intersection with every nonzero submodule of M/N).

Hence

(ii) If E is an essential submodule of M, and f : E → ∏_{I} L_{i} a subdirect decomposition of E, then there exists a subdirect decomposition g : M → ∏_{i} M_{i} of M, such that each M_{i} is an overmodule of L_{i} in which L_{i} is essential, and f is the restriction of g to E ⊆ M.

In particular, every locally Artinian module can be written as a subdirect product of locally Artinian modules with simple socles.

Proof. In the situation of (i), let N be maximal among submodules of M having trivial intersection with L. The triviality of this intersection means that L embeds in M/N, while the maximality condition makes the image of L essential therein. (If it were not essential, M/N would have a nonzero submodule M′ disjoint from the image of L, and the inverse image of M′ in M would contradict the maximality of N.)

In the situation of (ii), for each j ∈ I let K_{j} be the kernel of the composite E → ∏_{I} L_{i} → L_{j}. Applying statement (i) with M/K_{j} in the role of M, and E/K_{j} ≅ L_{j} in the role of L, we get an image M_{j} of M/K_{j}, and hence of M, in which L_{j} is embedded and is essential. Now since E is essential in M, every nonzero submodule M′ ⊆ M has nonzero intersection with E, and that intersection has nonzero projection to L_{i} for some i; so in particular, for that i, M′ has nonzero image in M_{i}. Since this is true for every M′, the map M → ∏_{I} M_{i} is one-to-one, and gives the desired subdirect decomposition.

To get the final assertion, note that the socle of a locally Artinian module is essential, and, being semisimple, can be written as a subdirect product (indeed, as a direct sum) of simple modules; so we can apply (ii) with E = socle(M) and the L_{i} simple. Each of the M_{i} in the resulting decomposition will have a simple essential submodule L_{i}, so that submodule must be its socle. □

We can now get the following result, showing that given a faithful module M over an Artinian ring, we can carve out of M a “small” faithful submodule, factor-module, or subfactor. Note that in the statement, the length of socle(R) as a bimodule may be less than its length as a left or right module. (For example, the full n × n matrix ring over a division ring is its own socle, and has length n as a right and as a left module, but length 1 as a bimodule. Similarly, in the R of §5.2, each of the direct summands x F and F y of socle(R) has length 2 as left and as right R-module, but length 1 as an R-bimodule; so socle(R) has length 4 on each side, but 2 as a bimodule.)

Proposition 17. Let R be a left Artinian ring, let n be the length of socle(R) as a bimodule (equivalently, as a 2-sided ideal), and let M be a faithful R-module. Then

(i) M has a submodule M′ which is again faithful over R, and satisfies length(M′/J(R) M′) ≤ n. (In particular, M′ is generated by ≤ n elements.)

(ii) M has a homomorphic image M′′ which is faithful over R, and satisfies length(socle(M′′)) ≤ n. (iii) M has a subfactor faithful over R satisfying both these inequalities.

Proof. To get (i), note that since R/J(R) is semisimple Artin, M/J(R)M can be written as a direct sum of simple modules L_{i} (i ∈ I) over that ring, and hence over R, so we can construct a generating family of submodules N_{i} ⊆ M related to these as in Lemma 15(ii). Since M = ∑_{i} N_{i} is faithful, and socle(R) has length n as a 2-sided ideal, the sum of some family of ≤ n of these submodules, say

(32) M′ = N_{i_{1}} + ⋯ + N_{i_{m}} where m ≤ n,

must have the property that M′ is annihilated by no nonzero subideal of socle(R). (Details: one chooses the N_{i_{j}} recursively; as long as N_{i_{1}}+⋯+N_{i_{j}} is annihilated by a nonzero subideal I ⊆ socle(R), one can choose an N_{i_{j+1}} which fails to be annihilated by I. The annihilators in socle(R) of successive sums M′ = N_{i_{1}}+⋯+N_{i_{j}} (j = 0, 1, …) form a strictly decreasing chain, so this chain must terminate after ≤ n steps.) Since an ideal of an Artinian ring having zero intersection with the socle is zero, M′ has zero annihilator, i.e., is faithful. Since each N_{i} satisfies length(N_{i}/J(R)N_{i}) = 1, we have length(M′/J(R)M′) ≤ m ≤ n.

Statement (ii) is proved in the analogous way from the final statement of Lemma 16, using images of M in products of finite subfamilies of the M_{i} in place of submodules of M generated by finite subfamilies of the N_{i}.

Statement (iii) follows by combining (i) and (ii). □

7. Acknowledgements

I am indebted to Luchezar Avramov for pointing me to Gulliksen’s [3, Lemma 2], to Clément de Seguins Pazzis for providing, at [6], the proof of the present version of Proposition 1, and to Pace Nielsen for many helpful comments on an earlier draft of this note.

References

[1] George M. Bergman, Commuting matrices, and modules over Artinian local rings, unpublished note, 14 pp., readable at

    /~gbergman/papers/unpub and at http://arxiv.org/abs/1309.0053 .

[2] David Eisenbud, Linear sections of determinantal varieties, Amer. J. Math. 110 (1988) 541–575. MR 89h:14041.

[3] Tor H. Gulliksen, On the length of faithful modules over Artinian local rings, Math. Scand. 31 (1972) 78-82. MR 47 #3370.

[4] T. Y. Lam, A first course in noncommutative rings, Springer GTM, v.131, 1991. MR 92f:16001.

[5] Kevin C. O’Meara, John Clark and Charles I. Vinsonhaler Advanced topics in linear algebra. Weaving matrix problems through the Weyr form, Oxford University Press, 2011. xxii + 400 pp. MR 2849857.

 [6] http://mathoverflow.net/questions/141378/is-this-lemma-in-elementary-linear-algebra-new .

University of California, Berkeley, CA 94720-3840, USA

   E-mail address: gbergman@math.berkeley.edu