Integral domain.

Question

Let k a field.

Let $ A = k+ x^2 k[x] $,

Show that A is integral domain and finite type.

An element of $A$ is $\alpha + x^2. f(x) $ where$ f\in k[x] $

I know that A is a subalgebra of k[x], Any hint for integral domain

Answer 1

Since $A\subset k[x]$ and since $k[x]$ is an integral domain, we can see that for all $a,b\in A$, $(ab = 0$ in $A) \Rightarrow (ab = 0$ in $k[x]) \Rightarrow (a=0$ or $b=0)$.