Definition:Minimal Condition

Minimal condition on subsets
Let $S$ be a set.

Let $F$ be a set of subsets of $S$.

Then $S$ satisfies the minimal condition on $F$ $F$, ordered by inclusion satisfies the minimal condition.

Minimal condition on submodules
Let $A$ be a commutative ring with unity.

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

Let $(D,\supseteq)$ be the set of submodules of $M$ ordered by inclusion.

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