# Ring isomorphic to $\mathbb{Z}[1/3]/(5)$?

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 $$R/I \cong S/(S ∩ I)$$.

• 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 $3x-1$? – André 3000 Nov 9 '18 at 0:11
• If I had to guess it would simplify to $\mathbb{F}_5$ because $R[x]/(x) \cong R$ – Mac Nov 9 '18 at 0:21
• 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! – André 3000 Nov 9 '18 at 1:24