Show that the quotient ring R/N has no non-zero nilpotent elements. An element $x$ in a ring $R$ is called nilpotent if $x^n=0$ for some $n\in \mathbb N$. Let $R$ be a commutative ring and $N=\{x\in R\mid \text{x is nilpotent}\}$.
(a) Show that $N$ is an ideal in $R$. 
(b) Show that the quotient ring $R/N$ has no non-zero nilptoent elements.
What's the steps to prove (a) and (b)?

The steps to prove (a) are simple:


*

*Take two elements $x,y\in N$ and show that $x+y\in N$

*Take an element $n\in N$ and an element $r\in R$ and show that $nr\in N$.


In order to prove (b), personally, I would take a nilpotent element in $R/N$ and show that it is the zero element.
