On envelopes and covers in the category of short exact sequences (Q2006681)

The author studies the relationship between pre-envelopes (pre-covers) in the category of \(R\)-modules with the category of exact short sequences (Section 2), giving sufficient and necessary conditions such as Theorem 2.1 and 2.2. Next, the author sets conditions on pre-envelopes (pre-covers) in the category of \(R\)-modules using certain class of modules (Theorem 2.6 and Proposition 2.7). As applications, in Section 3, the author gives equivalent conditions using (pre-)coverings and (pre-)envelopes to define: left perfect ring, right coherent ring (left), right semi-hereditary ring (left), and left Noetherian ring.