# Equivalence of Definitions of Artinian Module

## Theorem

The following definitions of the concept of Artinian Module are equivalent:

### Definition 1

$M$ is a Artinian module if and only if:

$M$ satisfies the descending chain condition.

### Definition 2

$M$ is a Artinian module if and only if:

$M$ satisfies the minimal condition.

## Proof

### Definition 1 iff Definition 2

Let $D$ be the set of all submodules of $M$.

We shall show that:

descending chain condition
minimal condition

with respect to $\struct {D, \supseteq}$ are equivalent.

This is nothing but:

ascending chain condition
maximal condition

with respect to $\struct {D, \subseteq}$ are equivalent.

The latter follows from Increasing Sequence in Ordered Set Terminates iff Maximal Element.

$\blacksquare$