On flatness of acts. (Q1423721)

Let \(S=G\dot\cup I\) be a monoid where \(G\) is a group and \(I\) an ideal of \(S\). It is proved that if an \(S\)-act is principally weakly flat, (weakly) flat, torsion free or satisfies condition \((P)\) or \((P_E)\) as an \(I^1\)-act, then it has these properties as an \(S\)-act as well. Examples are given showing that an \(S\)-act which is free, projective or strongly flat as an \(I^1\)-act is not necessarily free, projective or strongly flat as an \(S\)-act.