## Annihilator of a Submodule is a 2-Sided Ideal of a Ring

## Problem 410

Let $R$ be a ring with $1$ and let $M$ be a left $R$-module.

Let $S$ be a subset of $M$. The **annihilator** of $S$ in $R$ is the subset of the ring $R$ defined to be

\[\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.\]
(If $rx=0, r\in R, x\in S$, then we say $r$ **annihilates** $x$.)

Suppose that $N$ is a submodule of $M$. Then prove that the annihilator

\[\Ann_R(N)=\{ r\in R\mid rn=0 \text{ for all } n\in N\}\]
of $M$ in $R$ is a $2$-sided ideal of $R$.