Rees short exact sequences and preenvelopes (Q6610421)

Let \(S\) be a monoid. The authors examine commutative diagrams of Rees short exact sequences of \(S\)-acts, in the sense of [\textit{Y. Chen} and \textit{K. P. Shum}, Semigroup Forum 65, No. 1, 141--148 (2002; Zbl 1006.20050)].\N\NLet \(\mathcal{E}_S\) be a category with objects consisting of Rees short exact sequences of right \(S\)-acts. The morphisms from one Rees short exact sequence to another are determined by a triple of right \(S\)-act morphisms between the right \(S\)-acts in two sequences, whenever these \(S\)-morphisms form a commutative diagram of the Rees short exact sequences. The authors provide conditions on the morphisms of \(\mathcal{E}_S\) that ensure they are monomorphisms, not only when all morphisms of the triple are monomorphisms. The same applies to epimorphisms. Additionally, they provide conditions among certain sequences of \(S\)-acts that imply the exactness of one sequence based on the exactness of others. They present a theorem establishing a connection between the flatness property of a left \(S\)-act \(M\) and the exactness of the induced sequences by functors \(-\otimes M\). This result improves on Corollary 3.1.1 in [\textit{M. Jafari} et al., Math. Slovaca 69, No. 5, 1293--1301 (2019; Zbl 1483.20116)] by providing conditions covering both the necessary and sufficient aspects for exactness.\N\NAdditionally, the authors investigate and establish connections between preenvelopes in the category of central \(S\)-acts and preenvelopes in \(\mathcal{E}_S\) while they notably investigate preenvelopes in \(\mathcal{E}_S\). The same holds for procovers.\N\NAdditionally, the authors investigate and establish connections between preenvelopes in the category of central \(S\)-acts and preenvelopes in \(\mathcal{E}_S\). They also consider these connections for precovers.