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I am supposed to find a ring that is isomorphic to $\mathbb{Z}[1/3]/(5)$ using an isomorphism theorem. My guess is that $\mathbb{Z}[x]/(5,1+3x)$ is isomorphic to this and in turn I think that $\mathbb{F}_5[x]/(1+3x)$ is also isomorphic to this but I have no idea how to prove it other than I think I should use the isomorphism theorem if $S + I = R$, then $R/I \cong S/(S ∩ I)$.

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    $\begingroup$ You've done great so far! Now note that $3x+1 = 3(x+2)$, so we have the equality of ideals $(3x+1) = (x+2)$. Can you simplify $\mathbb{F}_5[x]/(x+2)$? Edit: Actually, shouldn't that be $3x-1$? $\endgroup$ – André 3000 Nov 9 '18 at 0:11
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    $\begingroup$ If I had to guess it would simplify to $\mathbb{F}_5$ because $R[x]/(x) \cong R$ $\endgroup$ – Mac Nov 9 '18 at 0:21
  • $\begingroup$ Yes, and more generally $R[x]/(x-a) \cong R$ using the map $x \mapsto a$ and the First Isomorphism Theorem. Feel free to write an answer if you think you've solved your question! $\endgroup$ – André 3000 Nov 9 '18 at 1:24

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