Consider the ring $\mathbb Z_4[x]$. Clearly the elements of the form $2f(x)$ are zero divisors.
1. Is it true that they are all the zero divisors? I mean is it true that if $p(x)$ is a zero divisor then it is of the form $$ 2f(x) $$ for some $f(x) \in \mathbb Z_4[x]$? In other words, is the set of zero-divisors exactly the ideal $(2)$?
I believe it is true, but I do not know how to prove it.
Secondly, the elements $1+g(x)$, with $g(x)$ zero divisors, are clearly units: $(1+g(x))^2=1$.
2. Is it true that they are all the units? I mean is it true that if $p(x)\in \mathbb Z_4[x]$ is a unit then it is of the form $$ 1+g(x) $$ for some zero divisor $g(x)$?