Definition:Maximal Condition on Submodules

Definition

Let $A$ be a commutative ring with unity.

Let $M$ be an $A$-module.

Let $\struct {D, \subseteq}$ be the set of submodules of $M$ ordered by the subset relation.

Then the hypothesis:

Every non-empty subset of $D$ has a maximal element

is called the maximal condition on submodules.

Also see

• Definition:Ascending Chain Condition
• Definition:Noetherian Module
• Increasing Sequence in Ordered Set Terminates iff Maximal Element

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