Endomorphism rings generated using small numbers of elements
Abstract
Let R be a ring, M a nonzero left R-module, X an infinite set, and E the endomorphism ring of the direct sum of copies of M indexed by X. Given two subrings S and S' of E, we will say that S is equivalent to S' if there exists a finite subset U of E such that the subring generated by S and U is equal to the subring generated by S' and U. We show that if M is simple and X is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of elements that take each copy of M to itself) fall into exactly two equivalence classes, with respect to the equivalence relation above. We also show that every countable subset of E is contained in a 2-generator subsemigroup of E.
Cite
@article{arxiv.math/0508637,
title = {Endomorphism rings generated using small numbers of elements},
author = {Zachary Mesyan},
journal= {arXiv preprint arXiv:math/0508637},
year = {2012}
}
Comments
12 pages. In the new version the main result has been slightly generalized, references have been added (particularly in connection with Corollary 2, which had been known before), and several of the proofs have been rewritten to improve clarity