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
*A*_{1}, *A*_{2}, *A*_{3}.
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. Back to list of unpublished notes.

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. Back to list of unpublished notes.

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 *P _{i}*
(1≤

20 pp., last updated 8 June, 2018. tex. dvi. pdf. ps. arXiv:1802.01712. Back to list of unpublished notes.

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*⊗_{k}*B*-module
*M*⊗* _{k}N* is again faithful.
If

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 Back to list of unpublished notes.

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. tex. dvi. ps. pdf. arXiv:1309.0053. Back to list of unpublished notes.

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.

(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* = [*x*_{1}, *L*] +
[*x*_{2}, *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. PDF file Back to list of unpublished notes.

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 PDF ps Back to list of unpublished notes.

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

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 TeX PDF ps Back to list of unpublished notes.

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

7 pp., 1990. PDF. Back to list of unpublished notes.

Taking *A* = *Z*_{2} and
letting **C** be the subcategory
of **Set**_{A} 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. PDF. ps. Back to list of unpublished notes.

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

8 pp., 1986. PDF. ps. Back to list of unpublished notes.

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.

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 Back to list of unpublished notes.

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.

4 pp., 1979_{}
PDF (digitized).
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14 pp., 1979_{}
PDF (digitized).
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6 pp., 1979._{}
PDF (digitized).
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42 pp., 1978?_{}
PDF (digitized).
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8 pp., 1978. PDF (digitized). Back to list of unpublished notes.

(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.

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.

By regarding a **Z**-graded ring as
(**Z**/*p***Z**)-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.

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

29 pp. + 1 page of addenda, ≤1973.
PDF (digitized)._{}
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12 pp., Winter 1971-1972. PDF (digitized). Back to list of unpublished notes.

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