A characterization of left simple and simple semigroups

Let S be a semigroup. Show that S is left simple if and only if Sa = S for all a \in S, and is simple if and only if SaS = S for all a \in S.


Recall that a semigroup is called left simple if the only left ideal is S itself.

Suppose S is left simple. Now S(Sa) \subseteq Sa, so that Sa is a left ideal. Hence Sa = S. Conversely, suppose Sa = S for every a \in S. Let L be a left ideal. Now L \neq \emptyset, so we have a \in L for some a. Then S = Sa \subseteq L, and thus L = S. So S is left simple.

Similarly, we can show that S is right simple if and only if aS = S for all a \in S.

Now suppose S is simple. (That is, it has no nontrivial ideals.) Let a \in S. Now S(SaS) \subseteq SaS and (SaS)S \subseteq SaS, so that SaS is an ideal. Hence SaS = S. Now suppose SaS = S for all a \in S. If I is a two-sided ideal, then we have a \in I for some a \in S. Now S = SaS \subseteq I, so that I = S. Hence S is simple.

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