# nLab principal ideal monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

A monoid (or semigroup) $M$ is a principal left ideal monoid if all left ideals of $M$ are left principal ideals in $M$. Similarily, $M$ is a principal right ideal monoid if all right ideals of $M$ are right principal ideals. Finally, $M$ is a principal ideal monoid if it is both a principal left ideal monoid and a principal right ideal monoid.

## Examples

A group is a principal ideal monoid $G$ whose only ideal is isomorphic to $G$ itself.

Last revised on May 21, 2021 at 18:24:25. See the history of this page for a list of all contributions to it.