## Torsion Submodule, Integral Domain, and Zero Divisors

## Problem 409

Let $R$ be a ring with $1$. An element of the $R$-module $M$ is called a **torsion element** if $rm=0$ for some nonzero element $r\in R$.

The set of torsion elements is denoted

\[\Tor(M)=\{m \in M \mid rm=0 \text{ for some nonzero} r\in R\}.\]

**(a)** Prove that if $R$ is an integral domain, then $\Tor(M)$ is a submodule of $M$.

(Remark: an integral domain is a commutative ring by definition.) In this case the submodule $\Tor(M)$ is called **torsion submodule** of $M$.

**(b)** Find an example of a ring $R$ and an $R$-module $M$ such that $\Tor(M)$ is not a submodule.

**(c)** If $R$ has nonzero zero divisors, then show that every nonzero $R$-module has nonzero torsion element.