This is again a question asked to me by this user. He apparently quit using MO due to a busy time in personal and professional life and resulting difficulties in spending time here with patience. I am taking the liberty to ask it myself(with permission) as I consider him and his questions to be of value.

Let $A$ be unitary ring(ie ring with identity), $a \in A$ be such that for all ring homomorphisms $f : A \rightarrow B$, $B$ a unitary non-zero ring, $f (a)$ is not a unit in $B$.

[a unit in a unitary ring is an element both right and left invertible].

Does it follow that $a$ is nilpotent?

[in particular, $f(a)$ is neither left, nor right invertible for all $f : A \rightarrow B \neq 0$.]

A weaker version may be, if it $a \in A$ is such that $f (a)$ does neither left nor right invertible for all $f : A \rightarrow B \neq 0$ imply that $a$ is a nilpotent element?