Argerami and Farenick have found conditions for the Injective envelope of a separable C∗-algebra to be a von Neumann algebra. In this paper, we introduce an equivalent version of this result by finding conditions for the G-Injective envelope of a separable G-C∗-algebra A to be a von Neumann algebra, when G is a discrete group acting on A.

There are remarkable relations between the graded homological dimensions and the ordinary homological dimensions. In this paper, we study the Injective dimension of a complex of graded modules and derive its some properties. In particular, we define the  dualizing complex for a graded ring and investigate its consequences.

In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of Injective objects (with respect to the class of embeddings in the category 𝑄,-TOP of 𝑄,-topological spaces) in the comma category 𝑄,-TOP/(𝑌, , 𝜎, ), when (𝑌, , 𝜎, ) is a stratified 𝑄,-topological space, with the help of their 𝑇, 0-reflection. Further, we have proved that for any 𝑄,-topological space (𝑌, , 𝜎, ), the existence of an Injective hull of ((𝑋, , 𝜏, ), 𝑓,) in the comma category 𝑄,-TOP/(𝑌, , 𝜎, ) is equivalent to the existence of an Injective hull of its 𝑇, 0-reflection ((˜,𝑋, , ˜,𝜏, ), ˜,𝑓,) in the comma category 𝑄,-TOP/(𝑌, ˜, , 𝜎, ˜,) (and in the comma category 𝑄,-TOP0/(𝑌, ˜, , 𝜎, ˜,), where 𝑄,-TOP0 denotes the category of 𝑇, 0-𝑄,-topological spaces).

Injectivity with respect to some subclasses of monomorphisms in a category has been studied. In this article, we define the notion of regular poprime monomorphism in the category of S-posets with a monotone action of a pomonoid S on them and study M-injectivity where M is a subclass of regular poprime monomorphisms. More ever, we characterize their behaviour considered constructions such as the product, coproduct, direct sum, pullbacks, and pushouts of S-posets. The main purpose of this paper is to find a relation between regular poprime Injective and poprime extensions of S-posets.

Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite Injective dimension, then R is Cohen-Macaulay. The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open: (GB) If R admits a finitely generated R-module of finite Gorenstein-Injective dimension, then R is Cohen-Macaulay. Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of Injective dimension ≤ 1. Levin and Vasconcelos indicate that if x m\m2 is a non-zerodivisor, then for every finitely generated R/xR-module M, there is id R M= id R/xr M+1. Using this fact, they construct a finitely generated R-module of finite Injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B). In this paper we study the Gorenstein Injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein Injective dimension has finite Injective dimension. We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein Injective dimension.

The Injective chromatic number $\chi_i(G)$ of a graph $G$ is the smallest number of colors required to color the vertices of $G$ such that any two vertices with a common neighbor are assigned distinct colors. The Mycielskian or Mycielski graph $\mu(G)$ of a graph $G$, introduced by Jan Mycielski in 1955 has the property that, these graphs have large chromatic number with small clique number. The generalized Mycielskian $\mu_m(G),m>0$ (also known as cones over graphs) are the natural generalizations of the Mycielski graphs. In this paper, sharp bounds are obtained for the Injective chromatic number of generalized Mycielskian of any graph $G$. Further, the Injective chromatic number of generalized Mycielskian of some special classes of graphs such as paths, cycles, complete graphs, and complete bipartite graphs are obtained.

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.

Let R be a ring. The class of SA-Injective right R-modules (SAIR) is introduced as a class of soc-Injective right R-modules. Let N be a right R-module. A right R-module M is said to be SA-N-Injective if every R-homomorphism from a semi-artinian submodule of N into M extends to N. A module M is called SA-njective, if M is SA-R-Injective. We characterize rings over which every right module is SA-Injective. Conditions under which the class SAIR is closed under uotient (resp. directsums, pure homomorphic images) are given. The definability of the class SAIR is studied. Finally, relations between SA-injectivity and certain generalizations of injectivity are given.