| course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
33 |
IB |
2017 |
Groups, Rings and Modules |
|
Paper 2, Section II, E |
2017 |
Let $R$ be a commutative ring.
(a) Let $N$ be the set of nilpotent elements of $R$, that is,
$$N=\left{r \in R \mid r^{n}=0 \text { for some } n \in \mathbb{N}\right}$$
Show that $N$ is an ideal of $R$.
(b) Assume $R$ is Noetherian and assume $S \subset R$ is a non-empty subset such that if $s, t \in S$, then $s t \in S$. Let $I$ be an ideal of $R$ disjoint from $S$. Show that there is a prime ideal $P$ of $R$ containing $I$ and disjoint from $S$.
(c) Again assume $R$ is Noetherian and let $N$ be as in part (a). Let $\mathcal{P}$ be the set of all prime ideals of $R$. Show that
$$N=\bigcap_{P \in \mathcal{P}} P$$