Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.