nLab principal ideal of a monoid

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.

Related concepts

• ideal of a monoid

• principal ideal monoid

• Bézout monoid

• GCD monoid

• unique factorization monoid

• group