Abstract Algebra Answers

Show that for a semisimple module M over any ring R, the following conditions are equivalent:
(1) M is finitely generated;
(2) M is noetherian;
(3) M is artinian;
(4) M is a finite direct sum of simple modules.

Let R be a right semisimple ring. For x, y ∈ R, show that Rx =Ry iff x = uy for some unit u ∈ U(R).

Let R be a (left) semisimple ring. Show that, for any right ideal I and any left ideal J in R, IJ = I ∩ J. If I, J, K are ideals in R, prove the following distributive law: I ∩ (J + K) = (I ∩ J) + (I ∩ K)

Let R be the (commutative) ring of all real-valued continuous functions on [0, 1]. Is R a semisimple ring?

Is Z2 ⊕ Z3 ⊕ Z5 ⊕• • • semisimple Z-module?

Is Z2 × Z2 × Z2 ו • • semisimple Z-module?

Is Z12 semisimple Z-module?

Is Z_6 semisimple Z-module?

Is Z_4 semisimple Z-module?

Is Q/Z semisimple Z-module?