Equivalence of Definitions of Artinian Module

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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.

$\blacksquare$