On Homological Classification of Monoids by Condition ( IP ) of Right S -Acts

In this paper, we introduce Condition [Formula: see text], which is an idempotent-based variant of Condition [Formula: see text] for acts over monoids, and develop connections with classical flatness properties. A classification of monoids will be given for which all (cyclic, monocyclic, Rees factor) acts satisfy Condition [Formula: see text]. A description of Rees factor acts satisfying Condition [Formula: see text] will be given by a new concept called quasi-completely left annihilating. Finally, the existence of [Formula: see text]-covers for cyclic acts will be studied via a new structure called an [Formula: see text]-decomposition.