# Answer to Question #16814 in Abstract Algebra for Melvin Henriksen

Question #16814

Let R be a ring with center C. Show that a right ideal A of R is an ideal if:

R/A is a cyclic left C-module.

R/A is a cyclic left C-module.

Expert's answer

Since

*S**⊇**C*, End(*S*(*R/A*)) is a subring of End(*C*(*R/A*)). Therefore, we have End(*C*(*R/A*)) is a commutative ring. Next, suppose "*R/A*is a cyclic left*C*-module " holds. Then*C*(*R/A*) can be identified with*C/I*for some ideal*I*of*C*. Then End(*C*(*R/A*))*∼*End(*C*(*C/I*))*∼*End(*C/I*(*C/I*))*∼**C/I*is a commutative ring, so we have End(*C*(*R/A*)) is a commutative ring. Finally, we have right ideal*A*of*R*is an ideal.Need a fast expert's response?

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