Show that the quotient ring R/N has no non-zero nilpotent elements.

Question

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)?

Answer 1

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.

Answer 2

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 $\widehat{x}$.