simple semigroup

Let $S$ be a semigroup. If $S$ has no ideals other than itself, then $S$ is said to be simple.

If $S$ has no left ideals [resp. right ideals] other than itself, then $S$ is said to be left simple [resp. right simple].

Right simple and left simple are stronger conditions than simple.

A semigroup $S$ is left simple if and only if $Sa=S$ for all $a\in S$.A semigroup is both left and right simple if and only if it is a group.

If $S$ has a zero element $\theta$, then $0=\{\theta \}$ is always an ideal of $S$, so $S$ is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker definition is required.

Let $S$ be a semigroup with a zero. Then $S$ is zero simple, or $0$-simple, if the following conditions hold:

• $S^2\ne 0$

• $S$ has no ideals except $0$ and $S$ itself

The condition $S^2=0$ really only eliminates one semigroup: the 2-element null semigroup. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.