Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

Details Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

2004-05-12

arxiv.org/abs/math/0405216v1

J. Algebra. 283 (2005), no.1, 364-366

Mathematics

Rings and Algebras

3 pages

Scientific paper

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.

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