Commutative ring is semisimple iff it's isomorphic to a finite direct product of fields.

A finite product of fields is clearly Artinian.

It's also then clearly Jacobson semisimple (since it has no nonzero nilpotent elements, and the radical is nilpotent.)

Since Jacobson radical zero and Artinian combine to make "semisimple," we're done.

One way is to show that every ideal of a finite direct product of fields is a direct summand. Can you see why?