Showing that two rings are not isomorphic

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SUMMARY

The rings R = ℤ/3 × ℤ/3 and S = (ℤ/3)[x]/(x²+1) are not isomorphic. The reasoning is based on the fact that S is isomorphic to (ℤ/3)[i], where i satisfies i² = 2. Since (ℤ/3)[i] is not isomorphic to ℤ/3 × ℤ/3, the conclusion is definitive that R and S cannot be isomorphic.

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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?

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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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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.
 
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