Monoids over which all flat left acts are regular (Q1921363)

Let \(S\) be a monoid. A left \(S\)-act \({_SA}\) is flat (strongly flat) if the functor \(A\otimes -\) preserves monomorphisms (pullbacks, respectively). A left \(S\)-act \(A\) is called regular if for any \(a\in A\) there exists a homomorphism \(f:Sa\to S\) such that \(f(a)a=a\). The main theorem states that the following conditions on a monoid \(S\) are equivalent: 1) All flat left \(S\)-acts are regular; 2) All weakly flat left \(S\)-acts are regular; 3) Every cyclic subact of every flat left \(S\)-act is strongly flat; 4) Every cyclic subact of every weakly flat left \(S\)-act is strongly flat; 5) Every element of \(S\) different from 1 is a left zero.