# Answer to Question #16811 in Abstract Algebra for Melvin Henriksen

Question #16811

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.

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.

Expert's answer

(2)

(1)

*⇒*(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**denote right multiplication by*_{r}*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*_{r}*ρ**=*_{r'}*ρ*_{r'}*ρ**. Applying the two sides of this equation to the coset 1 +*_{r}*A**∈**R/A*, we get*rr’*+*A*=*r'r*+*A*, which clearly implies (2).
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