# Tagged: submodule criteria

## Problem 416

Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain
$N_1 \subset N_2 \subset \cdots$ of submodules of $M$.
Prove that the union
$\cup_{i=1}^{\infty} N_i$ is a submodule of $M$.

## 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.