Example of a quasi nilpotent element which is not a nilpotent element

Question

Let $R$ be a ring with unity. An element $a\in R$ is said to be a quasi nilpotent element of $R$ if $1-ax$ is unit for all $x\in comm(a)$ where $comm(a)=\lbrace x \in R | ax=xa\rbrace $. It is obvious that all nilpotent elements are quasi nilpotent elements, but converse need not be true. I am having hard time finding a good example.

Answer 1

Let $R=\mathbb{R}[[x]]$ be the ring of formal power series with coefficients in $\mathbb{R}$. This is a commutative ring, and $a_0+\sum_{n=1}^{\infty}a_nx^n$ is a unit if and only if $a_0\neq 0$.

In this ring, $x$ is not nilpotent; but for any element $s\in \mathbb{R}[[x]]$, we have that the constant term of $1-sx$ is $1$, so $1-sx$ is a unit. Thus, $x$ is quasi-nilpotent but not nilpotent.