Ring with Infinitely Many Simple Modules

[gauss mouse]

Homework Statement

Give an example of a ring $R$ with infinitely many non-isomorphic simple modules.

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?

 

[mighty2000]

Your example is correct, but here is another example that may be more intuitive. Consider the ring of polynomials over a field F, denoted by F[x]. This ring has infinitely many non-isomorphic simple modules, namely the F[x]-modules F[x]/(p(x)) for any irreducible polynomial p(x) in F[x]. These modules are simple because the quotient ring F[x]/(p(x)) is a field, and they are non-isomorphic because they have different underlying fields.