Flatness and coproducts. (Q2502260)

The paper considers the relation of some kinds of flatness properties between right acts and their coproducts. It is proved that: For a monoid \(S\) and a right \(S\)-act \(A=\dot{\bigcup}_{i\in I}A_i\), where \(A_i\), \(i\in I\), are right \(S\)-acts, \(A\) is torsion free, principally weakly flat, weakly flat, flat, strongly flat, principally weakly homoflat, weakly homoflat, weakly pullback flat, pullback flat, principally weakly kernel flat, weakly kernel flat, translation kernel flat, satisfies conditions (P), (PF) and (PF') if and only if \(A_i\) has these properties for every \(i\in I\). Consequently, an act admits the flatness properties which are presented above if and only if its indecomposable components have the corresponding properties.