George M. Bergman - abstracts of unpublished notes

Some embedding results for associative algebras

Suppose we wish to embed an (associative) k-algebra  A  in a k-algebra  R  generated in some specified way; e.g., by two elements, or by copies of given k-algebras  A1A2A3.  Mal'cev, Shirshov, and Bokut' et al., have obtained sufficient conditions for such embeddings to exist.  We prove here some further results on this theme.

In particular, we merge the ideas of existing constructions based on two generating elements, and on three generating subalgebras, to get a construction using two generating subalgebras.

We pose some questions on how far these results can be strengthened.

10 pp., last revised 2 Nov., 2020. texpdfarXiv:2011.01448. Back to list of unpublished notes

Some questions for possible submission to the next Kourovka notebook

I meant to submit a few questions to the 19th edition of the Kourovka Notebook of unsolved problems in group theory, but it appeared in 2018, before I got around to doing so; so I decided to put together in advance a list of questions to submit to the next one.  In addition to the ones I'd had in mind, I looked through my past papers, and found quite a few that seem worth submitting.  I list them here, along with a few new ones.

6 pp., latest update 21 Nov., 2019.  pdfarXiv:1904.04298Back to list of unpublished notes

Some results on counting linearizations of posets

In §1 we consider a 3-tuple  S = (|S|, ;, E)  where  |S|  is a finite set,    a partial ordering on  |S|,  and  E  a set of unordered pairs of distinct members of  |S|,  and study, as a function of  n ≥ 0,  the number of maps  φ: |S| → {1,...,n}  which are both isotone with respect to the ordering  ≼,  and have the property that  φ(x) ≠ φ(y)  whenever  {x, y} ∈ E.  We prove a number-theoretic result about this function, and use it in §8 to recover a ring-theoretic identity of G. P. Hochschild.

In §2 we generalize a result of R. Stanley on the sign-imbalance of posets in which the lengths of all maximal chains have the same parity.

In §3-§6 we study the linearization-count and sign-imbalance of a lexicographic sum of  n  finite posets  Pi  (1≤in)  over an n-element poset  P0.  We note how to compute these values from the corresponding counts for the given posets  Pi,  and for a lexicographic sum over  P0  of chains of lengths  card(Pi).  This makes the behavior of lexicographic sums of chains over a finite poset  P0  of interest, and we obtain some general results on the linearization-count and sign-imbalance of these objects.

20 pp., last updated 8 June, 2018.  texdvipdfpsarXiv:1802.01712Back to list of unpublished notes

Tensor products of faithful modules

If  k  is a field,  A  and  B  k-algebras,   M   a faithful left A-module, and  N  a faithful left B-module, we recall the proof that the left  AkB-module  MkN  is again faithful.  If  k  is a general commutative ring, we note some conditions on  A, B, M  and  N  that do, and others that do not, imply the same conclusion.  Finally, we note a version of the main result that does not involve any algebra structures on  A  and  B

4 pp., last revised 18 Oct., 2016.  TeX file  Postscript file  DVI file  PDF file  arXiv:1610.05178. Back to list of unpublished notes

Commuting matrices, and modules over Artinian local rings

Gerstenhaber showed in 1961 that any commuting pair of  n×n  matrices over a field  k  generates a k-algebra  A  of k-dimension ≤ n.  A well-known example shows that the corresponding statement for 4 matrices is false.  The question for 3 matrices is open.

Gerstenhaber's result can be looked at as a statement about the relation between the length of a 2-generator finite-dimensional commutative k-algebra  A  and the lengths of faithful A-modules.  Wadsworth generalized this result to a larger class of commutative rings than those generated by two elements over a field.  We recover his generalization, using a slightly improved argument.

We then explore some examples, raise further questions, and make a bit of progress toward answering some of them.

12pp., last revised 1 Oct., 2013.  texdvipspdfarXiv:1309.0053. Back to list of unpublished notes

Addenda to "On common divisors of multinomial coefficients"

In the note to which this is an addendum, after proving a conjecture of Erdős and Szekeres on g.c.d.'s of two binomial coefficients, I obtain some restrictions on possible counterexamples to a conjecture of Wasserman's on when  k  k-nomial coefficients must have a common divisor > 1 (mostly for  k = 3).  That material was originally written up in a leisurely fashion.  I subsequently rewrote it to "cut out the fat", and moved the deleted material to this addendum, to which I later added one new result that I decided belonged in the same category.

Details:   In §1, we verify a computational result asserted without proof in that note, related to the possibility of strengthening the conjecture of Erdős and Szekeres proved there.  The remaining sections concern Wasserman's conjecture:  In §2 some basic concepts are recalled; in §3 we verify "by hand", to give a feel for the relevant considerations, that there is no counterexample to that conjecture with  k = 3  and  n = 78.  In §4 we strengthen counterexamples (to certain plausible generalizations of the conjecture) that were given in that note by proving that that, under a standard conjecture on prime values assumed by polynomials, there must be infinitely many such counterexamples for every  k.  In §5 we discuss how a certain argument in that note might be improved.  In §6 we note how the proofs of two results in that note can be modified to give estimates of different quantities from those estimated in the original results.

9 pp., March 2010.  PDF file  Back to list of unpublished notes

The condition  (∃ x1, ... , xn)  A = x1A + ... + xn A   in nonassociative algebras, and base-change

To any not-necessarily-associative finite-dimensional algebra  A  over a field  k  such that  AA = A,  i.e., such that every element may be written as a sum of products, we can associate the least integer  n  such that  A  can be written  x1 A+ ... + xn A  for  x1, ..., xn∈ A.  If  k  is infinite, this invariant is shown to be unchanged under extension of base field.  Counterexamples are noted if  k  is finite, or  A  infinite-dimensional.

(This was originally going to be applied in another paper, in combination with known results on when a Lie algebra  L  can be written L = [x1L] + [x2L].  But I and my coauthor of that paper learned of a more recently proved result, which could be used to get a stronger conclusion, albeit with the help of a more complicated argument.  Hence this write-up of the above relatively simple result was "orphaned".)

6 pp., 2009.  PDF file  Back to list of unpublished notes

Why do we have three sorts of adjunctions?

The three sorts of pairs of adjoint functors (covariant, contravariant right, and contravariant left) are explained as a double coset space  H \ D4 / K,  where  D4  is the dihedral group of order  8,  acting in a natural way on descriptions of the adjointness condition, and  H  and  K  are certain 2-element subgroups interchanging "trivially equivalent" adjoint pairs.

3 pp., 2008.  PDF  ps  Back to list of unpublished notes

Some empty inverse limits

A simplified proof is given of L. Henkin's result that every directed partially ordered set  I  of uncountable cofinality indexes a system of nonempty sets and surjective set maps with empty inverse limit.  Strengthening a result of N. Aronszajn rediscovered by G. Higman and A. H. Stone, it is then shown that a large class of such partially ordered sets index systems with these properties in which the sets occurring are all countable.  Examples are also given of ω1-indexed systems of nonzero cyclic modules over a ring  R  and surjective homomorphisms whose inverse limit is the zero module.  Several questions are posed.

I may revise and publish this; I'm not sure.

8 pp., 2005, revised 2011.  TeX  PDF  ps  Back to list of unpublished notes

Criteria for existence of semigroup homomorphisms and projective rank functions

Suppose  A,  S,  and  T  are semigroups,  : A → S  and  : A → T  semigroup homomorphisms, and  X  a generating set for  S.  We assume (1) that every element of  S  divides some element of  (),  (2) that  T  is cancellative,  (3) that  T  is power-cancellative (i.e,  xd = yd ⇒ x = y  for  d > 0),  and (4) a further technical condition, which in particular holds if  T  admits a semigroup ordering with the order-type of the natural numbers.  We show that there then exists a homomorphism  S → T  making a commuting triangle with  e  and  f  if and only if for every relation (x1, ... , xn ) = e(a)  holding in  S ,  with  x1, ... , xn ∈ X,  a ∈ A ,  and  w  a semigroup word, there exist  t1, ... , tn ∈ T  satisfying  (t1, ... , tn ) = (a).

This leads to an arithmetic criterion for the existence of integer-valued projective rank functions on rings.

7 pp., 1990.  PDFBack to list of unpublished notes

When  Cpt  has products that are not products in  C

The existence of a category  C  with the indicated property is not surprising; but the construction is weird enough to perhaps be of independent interest:  Given any abelian monoid  A,  we define a category  SetA  to have for its objects finite sets, each element of which is weighted'' by an element of  A,  and for its morphisms, set-maps such that the weight of every element of the codomain set is the sum of the weights of its inverse images in the domain set.

Taking  A = Z2  and letting  C  be the subcategory of  SetA  consisting of sets of total weight  0  then gives the desired example.  A simpler example, constructed with hindsight, is noted at the end.

2 pp., 1987.  PDFpsBack to list of unpublished notes

Von Neumann regular rings with tailor-made ideal lattices

An algebraic distributive lattice  L  in which the greatest element is compact can be represented as the ideal lattice of a Neumann regular algebra  R  over an arbitrary field  k  if either  (a) every element of  L  is a (possibly infinite) join of join-irreducible compact elements, i.e. if  L  is the  lattice  D(P)  of lower subsets of a partially ordered set  P,  or if  (b) every compact element of  L  is complemented, i.e. if  L  is the  lattice of ideals of a Boolean ring, or if  (c)  L  has only countably many compact elements.

In each case, the ring  R  is a direct limit of finite products of full matrix algebras over  k .

The results of this note were included in K. R. Goodearl and F. Wehrung, Representations of distributive semilattices in ideal lattices of various algebraic structures, Algebra Universalis 45 (2001) 71-102. MR2002g:06008.

For further strengthened results, see Giuseppe Baccella, Representation of partially ordered sets over von Neumann regular algebras. More prime, non-primitive regular rings, arXiv:2312.12194v3 (2024).

8 pp., 1986.  PDFpsBack to list of unpublished notes

Notes on Ferguson and Forcade's generalized Euclidean algorithm

The familiar Euclidean algorithm, applied to a pair of real numbers, will terminate if and only if they are commensurable; equivalently, if and only if they satisfy a linear relation with integer cofficients.  More generally, given an n-tuple of real numbers, the Euclidean algorithm can be used to detect the property that they are all commensurable, which is equivalent to their satisfying  n - 1  independent linear relations; but it does not give a way of determining whether they satisfy a smaller number of such relation (as, for instance  1,  √2,  and  1+√2  do).

An algorithm which will find such relations was described, but without motivation, by H. R. P. Ferguson and R. W. Forcade in Bull. A. M. S. in 1979.  In this note, I motivate a construction similar to theirs, and discuss a number of related questions.  It consists of a 51-page write-up, followed by 10 pages of corrections and addenda prepared after subsequent correspondence with Ferguson and Forcade.  I did not have time to learn enough about the literature in the area to prepare this note for publication.

61 pp., 1980.  PDF (digitized).  Back to list of unpublished notes

Homogeneous elements and prime ideals in Z-graded rings

The main result is that in a Z-graded ring, any 2-sided ideal that properly contains a prime ideal contains a nonzero homogeneous element.  This is a 1979 rewrite of an unpublished note referred to in a few places in the literature as "On Z-graded rings" (1973), of which I don't seem to have kept a copy.

4 pp., 1979.  PDF (digitized).  Back to list of unpublished notes

Wild automorphisms of free p.i. algebras, and some new identities

In part I (the first 12½ pp.), some large families of automorphisms of a generic d × d matrix algebra  k <x1, ..., xn>d  are constructed.  It is shown that when n = 2,  these are not all tame.  Some criteria for an endomorphism of  k <x1, ..., xn>  to be an automorphism are discussed.

In part II (the remaining pages), identities for d × d matrices are studied using the trick of diagonalizing one of a generic family such matrices.  Among the results obtained are the nonexistence of nontrivial identities whose total degree in the other indeterminates is  < d,  the existence of an essentially unique identity whose degree in those indeterminates is  d,  and the existence of elements centralizing the distinguished indeterminate  y  but not lying in  Z[y]   (Z  the center of the whole ring).

(I submitted this for publication, but the referee pointed out that some of the identities I obtained were known, and I did not have time to look into this and rewrite it.)

42 pp., 1979.  PDF (digitized).  Back to list of unpublished notes

A free epic subalgebra of a free associative algebra

It is shown that every free algebra in more than one indeterminate has a proper subalgebra which is also free, such that the inclusion of that subalgebra in the whole algebra is an epimorphism in the category of associative rings.

4 pp., 1979  PDF (digitized)Back to list of unpublished notes

The class of free subalgebras of a free associative algebra is not closed under taking unions of chains, or pairwise intersections

Examples are given establishing the results stated in the title, and the analogous results for free commutative associative algebras (polynomial rings).

14 pp., 1979  PDF (digitized)Back to list of unpublished notes

A derivation on a free algebra, whose kernel is a nonfree subalgebra.

It is shown that in the free algebra on 3 generators  x, y, z  over a field, the derivation given by  d(x) = xyx + xd(y) = - yxy - yd(z) = - x  has the asserted property.

6 pp., 1979.  PDF (digitized)Back to list of unpublished notes

The Lie algebra of vector fields on  Rn  satisfies polynomial identities.

After obtaining identities as referred to in the title, I learned that that result had been obtained by some Soviet mathematicians, and I stopped working on this write-up; so it was left in an unfinished and somewhat messy state.  It was later pointed out to me that one claim in my first theorem is incorrect, and I have crossed that out in the copy I digitized.  For more such background, see "p. 1 1/2".  There seems to be very little in the literature about the subject, so I am making this rough draft available. Incidentally, though motivated by the case of vector fields on  Rn,  the results actually concern Lie algebras of derivations on a general commutative ring.

42 pp., 1978?  PDF (digitized)Back to list of unpublished notes

A note on growth functions of algebras and semigroups

The main result is that any semigroup, or associative algebra over a field, with less than quadratic growth, has at most linear growth.  It has been incorporated into the literature, but from time to time people ask for the original note, so here it is.  (I worked this out at Oberwolfach, where I typed it up on a borrowed typewriter.)

8 pp., 1978.  PDF (digitized).  Back to list of unpublished notes

A note on Engel's Theorem

Theorem.  Let  T  be a set of endomorphisms of a finite-dimensional vector space  V  over a field  k.  Suppose that for all  s, t ∈ T  there exists  a  in the unital subalgebra of  End(V)  generated by  s  and  t  such that  st + as ∈ T,  and that all elements of  T  are nilpotent.  Then  Tn = 0  for some  n

(Engel's theorem is the case where  T  is a k-subspace of  End(V),  and  a = −1  for all  s  and  t,  making  T  a Lie algebra of linear operators.  Jacobson generalized this to the case where  a ∈ k  for all  s  and  t.)

3 pp., 1978.  PDF (digitized).  Back to list of unpublished notes

On generic relative global dimension 0

Given a ring homomorphism  K → R,  a condition stronger than "R  has left relative global dimension zero over  K" is described, which underlies many of the known examples of the latter condition.  (One of several equivalent conditions is that the universal derivation  R → ΩK(R)  split.)  Various classes of examples are noted.

In the final section (in connection with one of these examples, but independent of what precedes), I note a "Banach-Tarski paradox in field theory":  A field  E  can have two subfields  K  and  K′  such that  [E : K] = [E : K′] = ∞,  but such that  E = K + K′

22 pp.+2-page addendum, some time between 1976 and 1978.  PDF (digitized).  Back to list of unpublished notes

It is shown that if  R  is a (Z/nZ)-graded ring, then for each  a ∈ J(R),  every homogeneous component  ai  of  a  satisfies  n ai ∈ J(R).

By regarding a Z-graded ring as (Z/pZ)-graded for all primes  p,  it is deduced that the Jacobson radical of such a ring is a homogeneous ideal, and several classical results are recovered.

For  R  a ring graded by an arbitrary group  G,  the relation between the Jacobson radical of  R  as a graded ring, and as an ordinary ring, is examined.  A conjecture is made, which I believe has since been proved.

(I did not find time to put this in good form; I believe the main results have been incorporated into the literature.)

10 pp., 1975.  PDF (digitized).  Back to list of unpublished notes

A note on abelian categories — translating element-chasing proofs, and exact embedding in abelian groups

Given a theorem about abelian groups or R-modules which one proves by element-chasing arguments, one would like to get an analogous proof of the same result in an arbitrary abelian category.  In this note, we show that an appropriate reduced product of covariant Hom-functors gives the desired "elements".  More generally, this method, applied to a family of functors that are not necessarily Hom-functors, yields a sufficient condition for exact embeddability of an abelian category  A  in the category of abelian groups.

7 pp., 1974.  PDF (digitized).  Back to list of unpublished notes

Epimorphisms of Lie algebras

This note examines epimorphisms (in the category-theoretic sense), and the related concept of dominions of subalgebras (due to Isbell), in various categories of Lie algebras and p-Lie algebras over a field  k.  If no finite-dimensionality restriction is put on the algebras, and also for the category of finite-dimensional algebras when  char(k) ≠ 0,  the only epimorphisms are the surjective maps, and, in fact, every subalgebra is its own dominion.  In the finite-dimensional characteristic 0 case, nontrivial examples of epimorphisms occur, and are studied.  A page of addenda, based on comments I received when I sent out preprints, is included at the end.

29 pp. + 1 page of addenda, ≤1973.  PDF (digitized).  Back to list of unpublished notes

Functors from finite sets to finite sets

Given any functor  F,  covariant or contravariant, from the category  FS  of finite sets to itself, a function  f  from the natural numbers to the natural numbers is defined by  f ( |X| ) = |F(X)|.  This note investigates which functions from the natural numbers to the natural numbers arise in this way.  That question is answered completely for contravariant functors, and up to a possible added constant +1 for covariant functors.  References are given at the end to subsequent work in the literature, including a complete solution for the covariant case.

12 pp., Winter 1971-1972.  PDF (digitized).  Back to list of unpublished notes

The constructive theory of transcendental numbers, or the Liouville-Thue-Siegel-Mahler-Roth-Schneider Theorem

An expository write-up of a classical result on transcendental numbers, along with a quick survey of other results in the field, which I prepared as a grad student, in fulfillment of a requirement.

33 pp., 1967. PDF (digitized).  Back to list of unpublished notes