# Definition:Minimal Condition

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## Contents

## Definition

### Ordered set

Let $(P, \leq)$ be an ordered set.

Then $P$ satisfies the **minimal condition** if and only if it is **well-founded**:

- Every non-empty subset has a minimal element.

### 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 $F$** if and only if $F$, 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.