Definition:Minimal Condition

Definition

Ordered Set

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Then $S$ satisfies the minimal condition if and only if $\preccurlyeq$ 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 $S$.

Let $F$ be ordered by the subset relation.

Then $S$ satisfies the minimal condition on $F$ if and only if $F$ satisfies the minimal condition.

Minimal Condition on Submodules

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.

This article, or a section of it, needs explaining. In particular: Is there a good reason why the relation is reversed? This is at odds with how the subset relation is implemented in general. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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