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:
with respect to $\struct {D, \supseteq}$ are equivalent.
This is nothing but:
with respect to $\struct {D, \subseteq}$ are equivalent.
The latter follows from Increasing Sequence in Ordered Set Terminates iff Maximal Element.
$\blacksquare$