Question

Let R be a ring with center C. Show that a right ideal A of R is an ideal if: End(C(R/A)) is a commutative ring;

Answer 1

Since $S \supseteq C$, $\operatorname{End}(S(R/A))$ is a subring of $\operatorname{End}(C(R/A))$. Therefore, we have $\operatorname{End}(C(R/A))$ is a commutative ring. Then $C(R/A)$ can be identified with $C/I$ for some ideal $I$ of $C$. Then $\operatorname{End}(C(R/A)) \sim \operatorname{End}(C(C/I)) \sim \operatorname{End}(C/I(C/I)) \sim C/I$ is a commutative ring, so we have $\operatorname{End}(C(R/A))$ is a commutative ring. Finally, we have right ideal $A$ of $R$ is an ideal.