Rees short exact sequence and flatness properties (Q2003178)

Let \(S\) be a monoid with zero. The sequence \(\dots \to L_S\xrightarrow{f} M_S\xrightarrow{g} N_S\to \dots\) of right \(S\)-acts is called \textit{exact at \(M_S\)} if \(\ker g = (f(L)\times f(L))\cup 1_M\). A sequence \(0 \to L_S\to M_S\to N_S\to 0\) is called \textit{Rees short exact} if it is exact at \(L_S\), \(M_S\) and \(N_S\). This sequence is called \textit{left split} if there is an \(S\)-homomorphism \(f': M_S\to L_S\) such that \(f'f=1_{L_S}\). It is proved that if \(0 \to L_S\to M_S\to N_S\to 0\) is a Rees short exact sequence and \(L_S\) and \(N_S\) are torsion free (principally weakly flat, satisfying condition (E), projective, principally weakly injective, divisible) right \(S\)-acts, then \(M_S\) is also torsion free (principally weakly flat, satisfying condition (E), projective, principally weakly injective, divisible, respectively). The cases when \(L_S\) and \(N_S\) are weakly flat, flat or torsionless are proved under the assumption that the sequence is left split.