Endomorphism rings generated using small numbers of elements

Let R be a ring, M a nonzero left R ‐module, Ω an infinite set, and E = End R ( ⊕ Ω M ). Given two subrings S 1 , S 2 ⊆ E , write S 1 ≈ S 2 if there exists a finite subset U ⊆ E such that 〈 S 1 ∪ U 〉 = 〈 S 2 ∪ U 〉. We show that if M is simple and Ω is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of endomorphisms 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 .