I'm currently looking at prime elements in $\mathbb{Z}[[X]]$, so my question is whether $X$ or $1+X$ is prime in $\mathbb{Z}[[X]]$?
I personally believe that it is the case, because I can't think of any $a,b \in \mathbb{Z}[[X]]$ such that $X|ab$ but neither $X|a$ nor $X|b$ (or $1+X$ instead of $X$). But how do I show that? I thought I could compare this situation to showing that $X$ or $1+X$ is prime in $\mathbb{Z}[X]$. Back then, we basically used the fact that $\mathbb{Z}$ is a unique factorization domain and that $X$ and $1+X$ are irreducible in $\mathbb{Z}$. But how would one do that now?
The ring morphism $\mathbb Z[[X]]\to \mathbb Z:\sum_{j=0}^\infty a_jX^j\to a_0$ has as kernel the ideal $\langle X\rangle $, hence that ideal is prime since $\mathbb Z[[X]]/\langle X\rangle \cong \mathbb Z$ is a domain.
But the ideal $\langle X\rangle$ being prime is equivalent to the element $X$ being prime so that , yes, $X$ is prime.