Semisimple Rings are Noetherian

Question

I have seen the statement that if $R$ is a commutative ring and $R$ is a semisimple ring, then $R$ is Noetherian several times while reading through some algebra resources; however, I have never seen an explanation for it. The definition of a semisimple ring that I know is a ring that is the direct sum of minimal ideals. This definition alone doesn't imply that $R$ is Noetherian. Is there another way I should be thinking about it? Any help would be appreciated.

Answer 1

Hints:

• Every minimal ideal is a Noetherian $R$-module (why?).
• You can write $R$ as a finite direct sum of minimal ideals (why?).
• A finite direct sum of Noetherian modules is Noetherian.

Answer 2

You'd like to prove that every right ideal is finitely generated, but even more is true:

Every right ideal of a semisimple ring is principal!

Of course, you already know every right ideal is a summand, and a summand of $R_R$ is generated by an idempotent.