I am trying to prove that if $R$ is a ring and $I$ is a nilpotent ideal of $R$ and $I\neq N(R)$ then $R/I$ has a nonzero nilpotent element.
My attempt
Since $I$ nilpotent ideal of $R$ we have $I \subset N(R)$. So if $I\neq N(R)$ then $I$ is a strict subset of $N(R)$. So there is an $x \in N(R) \setminus I $.
Therefore $x+ I \in R/I $ and $x+ I \neq 0$. Since $x \in N(R)$, $x$ is nilpotent so there is an $n \in \mathbb{N}$ such that $x^n =0 $. So $(x+I)^n=x^n+I = 0 $ and so $x+I$ is a nonzero nilpotent element of $R/I$.
Is this proof ok?