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

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

Right simple and left simple are stronger conditions than simple.

A semigroup is left simple if and only if for all . A semigroup is both left and right simple if and only if it is a group.

If has a zero element , then is always an ideal of , so is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker definition is required.

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

• 
• has no ideals except 0 and itself

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