Definition:Minimal Condition/Submodule Ordering

Definition
Let $A$ be a commutative ring with unity.

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

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

Then the hypothesis:


 * Every non-empty subset of $D$ has a minimal element

is called the minimal condition on submodules.

Also see

 * Definition:Descending Chain Condition
 * Definition:Maximal Condition on Submodules