A quasi-injective weakly co-hopfian right module over a right Noetherian ring is characterized. Those right semi-Artinian rings and right FBN rings over which completely weakly co-hopfian right modules are precisely right modules of finite uniform dimension are characterized. In addition, some applications are provided.

In this paper, we introduce a proper generalization of that of Hop, an modules, called -Hop, an modules. A right R-module M is said to be -hopfian, if any surjective endomorphism of M has a -small kernel. Some basic characterizations of -modules are proved. We prove that a ring R is semisimple cosingular if and only if every R-module is -hopfian. Further, we prove that the -property is preserved under Morita equiva-lences. Some other properties of -hopfian modules are also obtained with examples.

The study of modules by properties of their endomorphisms has long been of interest. In this paper, we introduce the notion of jacobson weakly hopfian modules. It is shown that over a ring $R$, every projective (free) $R$-module is jacobson weakly hopfian if and only if $R$ has no nonzero semisimple projective $R$-module. Let $L$ be a module such that $L$ satisfies ascending chain conditions on jacobson-small submodules. Then it is shown that $L$ is jacobson weakly hopfian. Some basic characterizations of projective jacobson weakly hopfian modules are proved.

The present paper studies the hopfian and Co - hopfian objects in the category of rings and modules. Also the hopfian and Co - hopfian topological spaces, algebras and lattices are introduced, and the hopfian and CO - hopfian modules are charaterized. Furtheremore, the connection between the hopfian and Co - hopfian algebras and topological spaces is discussed.

Let $a$ be an ideal of a local ring $(R, m)$ and $M$ and  $N$ two finitely generated    $R$-modules. In this paper, we introduce the concept of Generalized formal local cohomology modules. We define $i$-th Generalized formal local cohomology module of $M$ and $N$ with respect to     $a$ by  $\mathfrak{F}_{a}^i(M,N) := \underset{n}{\varprojlim}H_m^i(M,N/{a}^{n}N )$ for $i\geq 0$. We prove several results concerning vanishing and finiteness properties of these modules.

Please click on PDF to view the abstract