# Answer to Question #16815 in Abstract Algebra for Melvin Henriksen

Question #16815

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

the factor group R/A is cyclic, or isomorphic to a subgroup of Q.

the factor group R/A is cyclic, or isomorphic to a subgroup of Q.

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, under " The factor group*R/A*is cyclic, or isomorphic to a subgroup of Q ", any Z-endomorphism of*R/A*is induced by multiplication by an integer or a rational number. Since End(*S*(*R/A*)) is a subring of End(Z(*R/A*)), we have that factor group*R/A*is cyclic, or isomorphic to a subgroup of QNeed a fast expert's response?

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