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)?

You should explain what you tried for answering both these questions. Namely, the first one (a) is quite easy once you go back to the definition of an ideal.
The second one should not take much longer: assume that $x \in R/N$ is nilpotent, let $\widehat{x} \in R$ be an antecedent of $x$, and look at what the assertion “$x$ is nilpotent” means for $\widehat{x}$.
