Answer to Question #16811 in Abstract Algebra for Melvin Henriksen

Question

S = IR(A) be the idealizer of A.
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.

Answer 1

(2) ⇒(1). Here, S = R since A is an ideal. Thus, End(S(R/A)) = End(R(R/A)) = End(R/A(R/A)) ∼ R/A, which is (by assumption) a commutative ring.
(1) ⇒(2). For r ∈ R, let ρ_rdenote right multiplication by r on R/A. This is meaningful since R/A is a right R-module, and we have ρ_r∈End(S(R/A)). Thus, for any r, r' ∈ R, (1) yields an equation ρ_rρ_r'= ρ_r'ρ_r. 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).