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.

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.

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.