Sequentially pure injectivity

Abstract

Injectivity with respect to pure monomorphisms was studied before in
categories of modules and categories of acts. In this paper we study this notion with respect to sequentially pure monomorphisms.
Although the Baer criterion for injectivity (weakly injectivity implies injectivity) is true for modules over a ring (with an identity), it is an open problem for acts over a semigroup S (with or without identity). In fact, we are not aware of any type of weak injectivity implying injectivity of S-acts, in general, other than the Skornjakov-Baer criterion, which says that injectivity with respect to subacts of cyclic acts implies injectivity with respect to all monomorphisms.
M.M. Ebrahimi and M. Mahmoudi in some papers have shown this criterion to be true for some special S.

Keywords: Sequentially pure, sequentially pure-injective.