# Showing that two rings are not isomorphic

## 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?

## 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?

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