since is reduced, the nilradical of is (0) and thus now see my solution to part 2) of the problem in this thread: http://www.mathhelpforum.com/math-he...-r-module.html

it'd help if instead of just posting your problem, you'd also tell us what you know! for example, do you know that every Artinian ring is Noetherian? (Hopkins-Levitzki theorem) or do you know

2. Now suppose is Artinian (need not be reduced). Prove that is Noetherian iff is Artinian.

about semisimple rings or composition series?