Finding simple modules of semisimple algebras Which of the following algebras are semisimple? For each, find all their simple modules:
• $A_1 := k[x]/(x^2 − 4)$ (assuming that $2 \ne 0 ∈ k);$
• $A_2 := \mathbb C[x]/(x^3 − x^2 + x − 1) \text{ and } A_2 := \mathbb Q[x]/(x^3 − x^2 + x − 1)$
• $A_3 := \mathbb F_2[x]/(x^2 + x + 1)$
Obviously if I can find simple modules then the algebras are semisimple I just don't know how to find them. I know that the simple A-modules are up to isomorphism the A-modules $K[X]/(h)$ where h is a irreducible polynomial that divides f.
Would it then be appropriate to factor each polynomial and express the algebras as these factorisations? 
 A: 
Obviously if I can find simple modules then the algebras are semisimple

Hm?  Every ring has simple modules... what would that prove about semisimplicity?  Do you mean showing the ring is a sum of simple ideals?

I know that the simple A-modules are up to isomorphism the A-modules []/(ℎ) where h is a irreducible polynomial that divides f.

Excellent! This is indeed useful.

Would it then be appropriate to factor each polynomial and express the algebras as these factorisations?

Yes.

Let me give you an example other than the ones you're working on that will help you get some intuition.
Consider $n=2^3*3^2*5$.  In $\mathbb Z$, there are three maximal ideals containing $n$: $(2)$, $(3)$ and $(5)$, and these correspond exactly to the maximal ideals of $\mathbb Z/n\mathbb Z$. Their intersection is $(2*3*5)=(30)$, so the Jacobson radical of $\mathbb Z/n\mathbb Z$ is not zero, and the ring is not semisimple. Do you see a suggestive relationship between $2^3*3^2*5$ and $2*3*5$?
Another thing that could have been noticed is that $2*3*5$ is obviously a nonzero nilpotent element, and in a commutative ring, the nilpotent elements all have to be contained in the Jacobson radical. Commutative semisimple rings of course, have a zero radical, so that is another way to see it is not semisimple.
I'm pretty sure the paragraphs above will enable you to immediately spot what the Jacobson radical of a quotient of a PID is in every case, including the ones you posed in your problem. Do you see the pattern?
