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:

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${S}^{2}\ne 0$

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$S$ has no ideals except $0$ and $S$ itself
The condition ${S}^{2}=0$ really only eliminates one semigroup: the 2element null semigroup. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.
Title  simple semigroup 

Canonical name  SimpleSemigroup 
Date of creation  20130322 13:05:59 
Last modified on  20130322 13:05:59 
Owner  mclase (549) 
Last modified by  mclase (549) 
Numerical id  7 
Author  mclase (549) 
Entry type  Definition 
Classification  msc 20M10 
Defines  simple 
Defines  zero simple 
Defines  right simple 
Defines  left simple 