Using First Isomorphism Theorem with quotient rings

  • #1

Homework Statement


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

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


Homework Equations


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


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

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

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

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