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.

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

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...

Three edges e1, e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph G = (V,E) is a coloring c of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then c(e1) 6= c(e3). The injective edge coloring number χ ′ i(G) is the minimum number of colors permitted in such a coloring. In this paper,...

Suppose A is a separable unital C(X)-algebra each fibre of which is isomorphic to the same strongly self-absorbing and K1-injective C∗-algebra D. We show that A and C(X) ⊗ D are isomorphic as C(X)-algebras provided the compact Hausdorff space X is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Let $$\mathfrak {M}(\Sigma )$$ be an open and connected subset of the space hyperbolic metrics on a closed orientable surface, {M}(M)$$ manifold dimension at least 3. We impose conditions M $${{\,\mathrm{\mathfrak {M}}\,}}(M)$$ , which are often satisfied when in have non-positive curvature. Under these conditions, data homotopy class maps from $$\Sigma $$ to allows us view )\times \mathfrak as...

Let $(R,fm,k)$ be a local Gorenstein ring of dimension $n$. Let $H_{I,J}^i(R)$ be the  local cohomology with respect to a pair of ideals $I,J$ and $c$ be the $inf{i|H_{I,J}^i(R)neq0}$. A pair of ideals $I, J$ is called cohomologically complete intersection if $H_{I,J}^i(R)=0$ for all $ineq c$. It is shown that, when $H_{I,J}^i(R)=0$ for all $ineq c$, (i) a minimal injective resolution of $H_{I,...

We investigate preserving of projectivity and injectivity by the object-wise tensor product of RC-modules, where C is a small category. In particular, let O(G,X) be the category of canonical orbits of a discrete group G, over a G-set X. We show that projectivity of RO(G,X)-modules is preserved by this tensor product. Moreover, if G is a finite group, X a finite G-set and R is an integral domain...