Using First Isomorphism Theorem with quotient rings

[PirateFan308]

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!

 

[mighty2000]

Thank you for your question. It seems like you are on the right track with using the First Isomorphism Theorem to prove the isomorphism between \mathbb{Z}_2[x]/I and \mathbb{Z}_2[x]/J. However, you do not need to find a specific ring S that both \mathbb{Z}/I and \mathbb{Z}/J are isomorphic to. Instead, you can use the given homomorphism \varphi: \mathbb{Z}_2[x] \rightarrow \mathbb{Z}_2[x] given by x \rightarrow x+1 to show that \mathbb{Z}_2[x]/I and \mathbb{Z}_2[x]/J are isomorphic to each other.

To do this, you can start by showing that \varphi is a surjective homomorphism with kernel I. Since \varphi is surjective, by the First Isomorphism Theorem, \mathbb{Z}_2[x]/I is isomorphic to \varphi(\mathbb{Z}_2[x]) = \{f(x+1) \mid f(x) \in \mathbb{Z}_2[x]\}. Can you see how this set is isomorphic to \mathbb{Z}_2[x]/J?

To show that \varphi has kernel I, you can use the fact that \varphi(x^2) = (x+1)^2 = x^2 + 2x + 1 = x^2 + 1 in \mathbb{Z}_2[x], as you mentioned. This means that x^2 \in ker \varphi. Can you think of any other elements in I that would also be in ker \varphi?

I hope this helps you get started on your proof. Good luck!