nLab principal ideal monoid

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.

Related concepts

• ideal of a monoid

• principal ideal of a monoid

• GCD monoid

• unique factorization monoid

• Bézout monoid

• group