George M. Bergman - abstracts of unpublished notes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
A1, A2, A3.
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.
tex.
pdf.
arXiv:2011.01448.
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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.
pdf.
arXiv:1904.04298.
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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≤i≤n)
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.
tex.
dvi.
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ps.
arXiv:1802.01712.
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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 A⊗kB-module
M⊗kN 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.
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arXiv:1610.05178.
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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.
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dvi.
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arXiv:1309.0053.
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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.
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The condition (∃ x1,
... , xn)
A = x1 A + ... + 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 = [x1, L] +
[x2, L].
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.
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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.
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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.
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Criteria for existence of semigroup homomorphisms
and projective rank functions
Suppose A, S, and T are
semigroups, e : A → S and
f : A → T semigroup homomorphisms,
and X a generating set for S.
We assume (1) that every element of S divides
some element of e (A ),
(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
w (x1, ... ,
xn ) = e(a) holding
in S , with
x1, ... , xn ∈ X,
a ∈ A , and w a semigroup word,
there exist
t1, ... , tn ∈ T
satisfying
w (t1, ... , tn ) =
f (a).
This leads to an arithmetic criterion for the existence of
integer-valued projective rank functions on rings.
7 pp., 1990.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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 + x,
d(y) = - yxy - y,
d(z) = - x has the asserted property.
6 pp., 1979.
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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?
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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.
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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.
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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.
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On Jacobson radicals of graded rings
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.
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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.
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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.
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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.
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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.
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