# "Simple ideal" of a semigroup

I am working through Mario Petrich's Introduction to Semigroups.

Lemma I.3.11 states:

If I is a simple ideal of a semigroup S, then I is the kernel of S

The problem is that he has not defined "simple ideal," nor do I find this term defined straightforwardly through a Google or Stackexchange search. He has defined a "simple semigroup" (no proper ideals), and a "kernel" of a semigroup (the intersection of all ideals). Based on the proof that follows, it seems like it means a simple subsemigroup of S that is also an ideal of S (I was able to follow the proof with that definition).

Here is the proof:

If $$J$$ is an ideal of $$S$$, then $$I \cap J \supseteq IJ$$ and hence $$I \cap J$$ is nonempty and is thus an ideal of $$S$$. But then $$I \cap J$$ is also an ideal of $$I$$, which by simplicity of $$I$$ implies that $$I \cap J = I$$ and thus $$I \subseteq J$$.

So, is this use of "simple ideal" an editorial error in the text? Or is there another definition of "simple ideal" that I should be aware of?