Monoids over which all flat cyclic right acts are strongly flat (Q1346306)

Let \(S\) be a monoid and consider right \(S\)-acts. The authors characterize monoids \(S\) over which all (torsion free, weakly flat, flat, \((P)\)) cyclic right \(S\)-acts are strongly flat or projective. Looking from the other side, the paper contains a characterization of right nil monoids (that is, for every \(x \in S\), \(x \neq 1\), there exists \(n \in \mathbb{N}\) such that \(x^ n\) is a right zero element of \(S\)). All results of this paper, together with corresponding results of former papers of various authors and open questions, are collected in a very convenient table at the end of the paper.