SCHWEITZER 1975 Problem 3

Let $S$ be a semigroup without proper two-sided ideals and suppose that for every $a,b \in S$ at least one of the products $ab$ and $ba$ is equal to one of the elements $a,b$ . Prove that either $ab=a$ for all $a,b \in S$ or $ab=b$ for all $a,b \in S$ .

L. Megyesi