Abstract Algebra Answers

Is Q semisimple Z-module?

Is Z semisimple Z-module?

Let {F_i : i ∈ I} be a family of fields. Show that the direct product R =(direct product)F_i is a semisimple ring iff the indexing set I is finite.

Can any ring be embedded as a subring of a left semisimple ring?

Is any subring of a left semisimple ring left semisimple?

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.

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.

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;

S = IR(A) be the idealizer of A. Show that (1), (2) are equivalent:
(1) A is a maximal right ideal and R/A is a cyclic left S-module.
(2) A is an ideal of R, and R/A is a division ring.

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.