Monoids over which all regular left acts are flat
A left act over a monoid is said to be regular if all its cyclic subacts are projective. A left \(S\)-act \(B\) is called flat if the functor \(- \otimes B\) preserves all monomorphisms, and \(B\) is called (principally) weak flat if \(-\otimes B\) preserves embeddings of (principal) right ideals of \(S\) into \(S\). \(B\) is called faithful if for arbitrary \(s, t \in S\), \(b \in B\) from \(sb = tb\) it follows \(s = t\). In the paper a characterization of monoids over which all regular left \(S\)-acts are flat (weak flat, principally weak flat, faithful) is given. This is an answer to some questions remaining open in the article of \textit{M. Kil'p} and \textit{U. Knauer} [J. Pure Appl. Algebra 46, 217-231 (1987; Zbl 0619.20048)].
scientific article; zbMATH DE number 736949