Showing that two rings are not isomorphic

[Mr Davis 97]

Homework Statement

Are the two rings $R = \mathbb{Z}/3 \times \mathbb{Z}/3$ and $S = (\mathbb{Z}/3)[x]/(x^2+1)$ isomorphic or not?

Homework Equations

The Attempt at a Solution

I think that they are not isomorphic. I think this because it seems to be the case that $(\mathbb{Z}/3)[x]/(x^2+1) \cong (\mathbb{Z}/3)[ i ]$, but that $(\mathbb{Z}/3)[ i ] \not \cong \mathbb{Z}/3 \times \mathbb{Z}/3$. Am I on the right track or is this wrong?

 

[fresh_42]

Sounds good. You have to explain what $i$ is, namely $i^2=2$ and perhaps you can show what $\mathbb{Z}_3$ is isomorphic to.