Michael Kinyon

Here is what I just put under "Professional Service" in my annual self-evaluation:

"I wrote about a dozen referee reports last year for various journals. I stopped keeping careful records of these because what is even the point?"

Ganesan's Theorem: If R is a commutative ring with exactly n > 0 zero divisors, then |R| ≤ (n+1)^2.

(Conventions: R does not necessarily have a unity; 0 itself is not a zero divisor.)

Proof: Let a_0 = 0, let a_1,...,a_n be the n zero divisors, and set a := a_1. Let b ≠ 0 be such that ab = 0. For each x in R, (xa)b = 0, and thus xa = a_i for some i = 0,...,n. For each i, let A_i = {x | xa=a_i }.

Now suppose |R| ≥ (n+1)^2+1 = n(n+1)+(n+2). By the pigeonhole principle, some A_i has at least n+2 elements, say, r_1,...,r_{n+2}. Then the n+1 elements r_1-r_2,...,r_1-r_{n+1} are nonzero and distinct, and satisfy a(r_1-r_i)=0 for each i. This contradicts the assumption that there are exactly n zero divisors. Therefore |R|≤(n+1)^2. QED

This is not Ganesan's proof, which, although easy, is not as elementary.

Commutativity isn't important; the proof actually shows that a not necessarily commutative ring R with exactly n>0 *left* zero divisors has order no more than (n+1)^2.

I can't take 100% credit for the proof. The basic idea for n=1 and n=2 appeared in a Quora answer by computer scientist David Ash in response to a question asking if there are noncommutative rings with unity with exactly two zero divisors. (Answer: no, because there are no noncommutative unital rings of order less than 8.) I noticed the connection of Ash's argument to Ganesan's Theorem and went from there.

"Michael, did this researcher, whose name and affiliation are printed so clearly, coauthor this paper with you?"

No, ResearchGate, you're dreaming, go back to sleep.

A letter to Taylor & Francis from (most of) what was the editorial board of Communications in Algebra. The header is by Jim Coykendall. The managing editor to whom Jim refers is Scott Chapman, who was apparently summarily dismissed by T&F.

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Math friends: Last evening just before 10pm, approximately 80% of the editorial board of Communications in Algebra resigned (the joint letter of resignation is below) and it appears that at least another 10% will step down independently. I would advise caution when considering this journal (possibly other Taylor and Francis journals?) until such time as it gets back on an even keel. As of this post (1 business day after the resignation), all of the resigning editors are still in the editorial system and none have been contacted. Additionally, the managing editor position has been vacant for a week and there has been no replacement (and no visible movement in that direction); the papers appear to be just piling up. If this situation is coupled with multiple full reviews, I would not submit any paper there for which a timely decision is required.
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To Whom It May Concern:

We as editorial board members at Communications in Algebra are sending this notification of our resignation from the board. This letter is being written to explain our position. We note at the outset that a number of the signatories are willing to finish their currently assigned queue if requested by Taylor and Francis.

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@tao@mathstodon.xyz On BlueSky, Kevin ( @xenaproject@mathstodon.xyz ) said you were collecting proofs of "650 implies 448" (or really, "650 implies xy=x"). I found a 27 step Prover9 proof and have just finished "humanizing" it. I did not look at the Vampire proof, but instead started from scratch, using methodology described in [1]. I'll just email you the LaTeX'ed PDF and the Prover9 proof itself. I don't speak Lean-ish so I can't do that conversion for you, but if someone wants to take it on, it's fine with me.

[1] M. Kinyon, Proof simplification and automated theorem proving, Philos. Trans. Roy. Soc. A, 377 (2019), no. 2140, 20180034, 9 pp.

arXiv version: https://arxiv.org/abs/1808.04251