# nLab principal ideal of a monoid

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

Given a monoid (or semigroup) $M$ and an element $a \in M$, a left principal ideal in $M$ is a subset $M a$ of $M$ such that for all $m \in M$, $m a \in M a$. Similarly, a right principal ideal in $M$ is a subset $a M$ of $M$ such that for all $m \in M$, $a m \in a M$. Finally, a two-sided principal ideal, or simply principal ideal, in $M$ is a subset $\langle a \rangle$ that is both a left ideal and a right ideal.

Last revised on May 21, 2021 at 18:23:57. See the history of this page for a list of all contributions to it.