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, the notion of Injectivity with respect to order dense embeddings in the category ofS-posets, posets with a monotone action of a pomonoidS on them, is studied. We give a criterion, like the Baer condition for Injectivity of modules, or Skornjakov criterion for Injectivity ofS-sets, for the order dense Injectivity. Also, we consider such Injectivity forS itself, and its order dense ideals. Further, we define and study some kinds of weak Injectivity with respect to order dense embeddings, consider their relations with order dense Injectivity. Also investigate if these kinds of Injectivity are preserved or reflected by products, coproducts, and direct sums ofS-posets.

In this article, inspiring with a result due to O.A.S. Karamzadeh, we examine the $\prod_{i\in I} R_i/\oplus_{i\in I} R_i$, where $\{R_i\}_{i\in I}$ is an infinite family of rings. We observe that they are not self-injective on either side. In some important cases they are however $\aleph_0$-self-injective. Along this line, we study the interconnection between regularity(in the sense of von Neumann), Injectivity and $\aleph_0$-Injectivity.

In this paper, the notion of C_v-Injectivity of monoid acts is studied. The behavior of C_v-Injectivity with respect to products, coproducts and direct sums is investigated. We characterize monoids over which all acts are C_v-injective and show that any act over such monoids is vitally injective. The class of acts satisfying C_v-Injectivity is determined. By means of the notion of C_v-Injectivity, we find conditions over which any projective act is injective. Also, it is shown that on a left reversible monoid, any C_v-injective act with zero is C-injective. We prove that if any C_v-injective act on monoid S is vital injective, then monoid S is left leversible. Using the concept of vitally injective envelope, a class of C_v-injective acts is obtained. Moreover, we consider the notion of M_v-Injectivity as a generalization of CC-Injectivity and study when the factor act of an M_v-injective act is M_v-injective. As a result, we show that any factor of a principally weakly vital injective act is principally weakly vital injective.

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.