Hereditary endomorphism monoids of projective acts (Q755907)

A monoid S is called left (semi)hereditary if all (finitely generated) left ideals of S are projective. Let P be a projective left S-act with at least two generators \(P=\cup_{i\in I}Se_ i\), \(| I| \geq 2\). In the paper it is shown that the endomorphism monoid \(End_ SP\) is (semi)hereditary if and only if \(P=Se_ 1\cup Se_ 2\) (i.e. \(| I| =2)\) and \(e_ 1Se_ 1\), \(e_ 2Se_ 2\) are groups. Note that in this case the groups \(e_ 1Se_ 1\), \(e_ 2Se_ 2\) are isomorphic.