On multiplicative semigroups of von Neumann regular rings (Q1320555)

A semigroup \(S\) is: 1) left absolutely flat if any left \(S^ 1\)-set is flat; 2) strongly left reversible if for any \(x,y \in M\) there exists \(z \in S^ 1\) such that \(zx = x\) and \(zy \in M \cap yM\). \(S\) is right absolutely flat (strongly reversible) if it is both left and right absolutely flat (strongly reversible). It is proved that \(R\) is a von Neumann regular ring if and only if the multiplicative semigroup of \(R\) is strongly reversible (or, equivalently, absolutely flat).