Answer to Question #16811 in Abstract Algebra for Melvin Henriksen
Show that (1), (2) are equivalent:
(1) End(S(R/A)) is a commutative ring.
(2) A is an ideal of R, and R/A is a commutative ring.
(1) ⇒(2). For r ∈ R, let ρrdenote right multiplication by r on R/A. This is meaningful since R/A is a right R-module, and we have ρr∈End(S(R/A)). Thus, for any r, r' ∈ R, (1) yields an equation ρrρr'= ρr'ρr. Applying the two sides of this equation to the coset 1 + A ∈ R/A, we get rr’ + A = r'r + A, which clearly implies (2).
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