Let $K$ be a field. I'd like to show that in $R=\{\sum a_i X^i\in K[X]\mid a_1=0\}$, the element $X^2$ is irreducible, but not prime.
Irreducibility is checked easily, but I can't see why it's not prime.
$X^2$ divides every polynomial with constant term $= 0$, but none of the nonzero ones. Thus, I have to multiply 2 polynomials with constant nonzero terms, but then the resulting polynomial also has constant term which isn't zero, so $X^2$ doesn't divide it?
Hint $\ X^2\mid\, X^3\cdot X^3,\:$ but $\, X^2\nmid X^3\,$ since $\,X^3/X^2 = X\not\in R$