# Ring with Infinitely Many Simple Modules

1. Apr 10, 2013

### gauss mouse

1. The problem statement, all variables and given/known data
Give an example of a ring $R$ with infinitely many non-isomorphic simple modules.
3. The attempt at a solution
I was thinking of setting
$R=\mathbb{Z}_{p_1}\times \mathbb{Z}_{p_2}\times \mathbb{Z}_{p_3}\times \cdots$
where $p_1,p_2,p_3,\ldots$ is an infinite increasing list of distinct prime numbers. Then each $\mathbb{Z}_{p_i}$ is an ideal of the ring and each $\mathbb{Z}_{p_i}$ is in fact simple because it is generated by any non-zero element.

Is this correct? Can anybody think of another (possibly better) example?