Equalizers and flatness properties of acts. II. (Q1434119)

[For part I see Commun. Algebra 30, No. 3, 1475-1498 (2002; Zbl 1002.20044).] A monoid \(S\) is said to be `right absolutely weakly equalizer-flat' if every right \(S\)-act is weakly equalizer-flat. An act \(A_S\) is called `torsion-free' if \(ac=a'c\Rightarrow a=a'\) whenever \(a,a'\in A\) and \(c\in S\) is right cancellable. It is proved that a monoid \(S\) where \(sx=sy\), \(s,x,y\in S\), implies the existence of a central idempotent \(f\in S\) such that \(sf=s\) and \(fx=fy\), is right absolutely weakly equalizer-flat. It follows that every right PP monoid with central idempotents is right absolutely equalizer-flat. It also has been shown that a torsion-free \(S\)-act need not be strongly torsion-free.