Principal Ideal in a Monoid

Let $M$ be a Monoid, and $x:M$.

• The principal left ideal generated by $x$ is the Left Ideal $\left\{\;a:M\right\}$,
• The principal left ideal generated by $x$ is the Right Ideal $\left\{\;a:M\right\}$,
• The principal two-sided ideal generated by $x$ is the Two-Sided Ideal $\left\{\;a:M\right\}$

Properties

• If $M$ is a Commutative Monoid, then all the above ideals coincide. This is not the case for a Commutative Semigroup; in general, ideals in semigroups are a rather cursed concept, and should be avoided.

• Principal ideals are closely linked with Divisibility; the principal left/right ideals of $x$ are essentially the subsets of $M$ of elements that are right/left divisible by $x$.