Isomorphism from $\Bbb Z_{2}[x]/(x^2+x)$ to $\Bbb Z_{2}$x$\Bbb Z_{2}$

Question

Is there an isomorphism from $\Bbb Z_{2}[x]/(x^2+x)$ to $\Bbb Z_{2}$x$\Bbb Z_{2}$?

I looked at $\phi$ where $\\$

$\phi(0)=(0,0)$

$\phi(1)=(1,1)$

$\phi(x)=(1,0)$

$\phi(x+1)=(0,1)$

and so far it seems that $\phi$ is a bijective homomorphism. Is this right or am I missing something?

Answer 1

Consider the morphism $f:\mathbb{Z}_2[x]\rightarrow \mathbb{Z}_2\times\mathbb{Z}_2$ defined by $f(1)=(1,1), f(x)= (0,1)$, $f(x(x+1)=(0,1)((1,1)+(0,1)=(0,1)(1,0)=0$ so $f$ factors by $\bar f:\mathbb{Z}_2/(x^2+x)\rightarrow \mathbb{Z}_2\times \mathbb{Z}_2$. If $p\in\mathbb{Z}_2[x]$, the Euclidean division of $p$ by $x^2+x$ gives $p(x)=q(x^2+x)+ax+b$, $f(p)=a(0,1)+(b,b)=(b,a+b)=0$ i.e $a=b=0$, this implies that $\bar f$ is injective, since $\bar f $ is trivially surjective, we deduce that $\bar f$ is an isomorphism.