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I am working on this question and I was wondering if I was on the right track.

It states: "Let $A$ be a ring. Prove that $A[x]/\langle x\rangle$ is isomorphic to $A$.

So am I on the right track in saying that I must check to see if the sum of $A[x]/\langle x\rangle$ goes to the sum of $A$, and the product of $A[x]/\langle x\rangle$ goes to the product of $A$?

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    @Neal Thanks Neal!2012-12-13

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Define $f: A[x] \to A$ as $p(x) \mapsto p(0)$. The kernel of this map is $\langle x \rangle$. Hence by the first isomorphism theorem $A[x]/ \langle x \rangle \cong A$.

It remains to be verified that $f$ is indeed a ring homomorphism that is, that the "product goes to the product" and the "sum goes to the sum" as you say, so you are on the right track. You should also convince yourself that $\mathrm{ker}f = \langle x \rangle$ and $\mathrm{im}f = A$.