For the subclasses M1 and M2 of monomorphisms in a concrete category C, if M2 Í M1, then M1-injectivity implies M2-injectivity. The Baer type criterion is about the converse of this fact. In this paper, we apply injectivity to the classes of C-dense, C-closed monomorphisms. The concept of quasi injectivity is also introduced here to investigte the Baer type criterion for these notions.

This paper explores generalized hyper S-acts (GHS-acts) over a hypermonoid S as generalizations of monoid acts within the context of algebraic hyperstructures. Specifically, we extend the definition of C-injectivity to GHS-acts and investigate their internal and homological properties. It is established that for being GHS-injectivity of GHS-acts with a fixed element, it suffices to consider allinclusions from cyclic GHS-subacts into indecomposable ones. Then we introducenew concepts known as semi-injectivity and semi-C-injectivity. By providing examples, we demonstrate that injectivity and semi-injectivity (C-injectivity and semiC-injectivity) are different concepts for GHS-acts, whereas they are the same in the context of acts over monoids. It is also shown that all pure GHS-acts are injective if and only if all pure cyclic GHS-acts are C-injective. Furthermore, we establish an equivalent condition on a hypermonoid S such that all quotients of SS exhibit semi-injectivity. Finally, we derive an equivalent condition for a hypermonoid to beclassified as semi-injective.

This article is a continuation of E. Momtahan’s work in which he studied the literature via countable injectivity. Along this line, N0-Ikeda-Nakayama rings have been studied and among other things, new characterizations of basically disconnected and extremally disconnected spaces have been given. Furthermore we will observe that a boolean ring is N0-self-injective if and only if it is an N0-IN ring.

We introduce the notion of approximately injectivity for Banach modules. For a locally compact group G, we study the approximately injectivity of Banach L1 (G)- modules.

In this paper some properties of weak regular injectivity for S-posets, where S is a pomonoid, are studied. The behaviour of different kinds of weak regular injectivity with products, coproducts and direct sums is considered. Also, some characterizations of pomonoids over which all S-posets are of some kind of weakly regular injective are obtained. Further, we give some Baer conditions which state the relation among some kinds of weak regular injectivity.