Answer to Question #16815 in Abstract Algebra for Melvin Henriksen

Question

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.

Answer 1

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 Q