In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular,...

A New generalization of injective semimodule has been presented in this working paper. An Ȑ-semimodule Ӎ is called almost self-injective, if Ӎ-injective semimodule. Some properties notion have presented. The relationship concept to some concepts also clarified as End (Ȑ) indecomposable self-injective semimodules, Rad related notions it studied.

In terms of the duality property of injective preenvelopes and flat precovers, we get an equivalent characterization of left Noetherian rings. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of RR. Then we get that the injective envelope of RR is (Gor...

This note investigates modules having quasi-injective and duo submodules. We introduce a new generalization of C_1-module. The main method that was adopted in this is how to obtain submodule N M the characteristic Quasi-injective. investigate relationship between pseudo-injective module Quasi-injective property Finally, we anti-hopfian module.

The injective chromatic number χi(G) [5] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. In this paper we define injective chromatic sum and injective strength of a graph and obtain the injective chromatic sum of complete graph, paths, cycles, wheel graph and complete bipartite graph. We a...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...

Characterisations of those separable C∗-algebras that have type I injective envelopes or W∗-algebra injective envelopes are presented. An operator system I is injective if for every inclusion E ⊂ F of operator systems each completely positive linear map ω : E → I has a completely positive extension to F . An injective envelope of an operator system E is an injective operator system I such that ...