Consider $\varphi :\mathbb Z_2[X] \to R$ where $R$ is ring:
$R:= \mathbb Z_2 \times \mathbb Z_2$
$(a,b) \times_R (c,d):= (ac, ad+bc)$
and the function is defined by $\varphi(\sum a_iX^i)= (a_0, a_1)$
I need to show that is a ring homomorphism, injective or surjective.
I tried showing that:
$f(a+_Sb)= f(a)+_R f(b)$ and $f(a\times_S b)= f(a)\times_R f(b)$
($+_S$ and $\times_S$ are the operators for $\mathbb Z_2[X]$, standard polynomial multiplication and addition)
The addition I could show to be valid but I think I might be doing something wrong for multiplication as its not adding up. Also how do I show if it is injective or surjective?