# Using First Isomorphism Theorem with quotient rings

## Homework Statement

Try to apply the First Isomorphism Theorem by starting with a homomorphism from a polynomial ring $R[x]$ to some other ring $S$.

Let $I = \mathbb{Z}_2[x]x^2$ and $J = \mathbb{Z}_2[x](x^2+1)$. Prove that $\mathbb{Z}_2[x]/I$ is isomorphic to $\mathbb{Z}/J$ by using the homomorphism $\mathbb{Z}_2[x] \rightarrow \mathbb{Z}_2[x]$ given by $x \rightarrow x+1$.

## Homework Equations

First isomorphism theorem states that if $\varphi: R \rightarrow S$ is a homomorphism then $R / ker \varphi$ is isomorphic to Image$\varphi$

## The Attempt at a Solution

The only thing that I was able to think of was to use the First Isomorphism Theorem twice to find some ring S that both $\mathbb{Z}/I$ and $\mathbb{Z}/J$ are isomorphic (hence making them isomorphic to each other), but I am completely stumped about which ring S I should use. I'm also confused about how the homomorphism given will help?

I attempted to find $kernel (\varphi)$, but had no luck as I'm confused about how which functions would be s.t. $f(x+1)=0$ in $\mathbb{Z}_2[x]$?

I did notice that $\varphi (x^2+1) = (x+1)^2+1 = x^2 + 2x + 1 + 1 = x^2$ in $\mathbb{Z}_2[x]$. Similarly, $\varphi (x^2) = (x+1)^2 = x^2 + 2x + 1 = x^2 + 1$, but I'm not sure how exactly this could help.

Any hint on where to get started would be greatly appreciated! Thanks!