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.