Quotienting out nilpotent elements
Let
Let
View code
Let $R$ be a commutative ring with $1$, and $N$ the ideal
\[
N=\{a\in R\,\mid\, a^n=0\text{ for some }n\}.
\]
Let $[b]$ be the image of $b\in R$ in $R/N$. Prove that if $[a]\in R/N$ and $[a]^m=0$ then $[a]=[0]$.