The purpose of this paper is to introduce the concepts of lacunary STRONGLYDm-summable andlacunary Dm -statistically convergent of GENERALIZED difference sequences of fuzzy numbers. Some results are obtained related to these concepts.

An R -module M is called STRONGLY noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) anR -module M is STRONGLY noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring Ris Artinian hereditary serial if and only if the class of injective modules coincides with the class of (STRONGLY) noncosingular R -modules; (3) absolutely coneat modules are STRONGLY noncosingular if and only if R is a right max ring and injective modules are STRONGLY noncosingular; (4) a commutative ring R is semisimple if and only if the class of injective modules coincides with the class of STRONGLY noncosingular R -modules.

Blumberg spaces has been introduced by H. Blumberg in 1922.In this article, we introduce STRONGLY Blumberg spaces and some of their properties will be investigated. We also consider important subspaces of STRONGLY Blumberg spaces such as Baire spaces. In particular, a generalization of White's theorem is considered.

Let M be a class of (mono) morphisms in a category A. To study mathematical notions, such as injectivity, tensor products and flatness, one needs to have some categorical and algebraic in-formation about the pair (A, M). In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and Msd to be the class of STRONGLYs-dense monomorphisms and study the categorical properties, such as limits and colimits, of this class.

In this paper we extend the concepts of two sided ideal and right quasi-duo ring. These ideals and rings are called totally fully in-variant and right STRONGLY quasi-duo, respectively. Right STRONGLY quasi-duo rings are always right quasi-duo. We investigate the properties of these rings and ideals and show among other things that right STRONGLY quasi-duo rings are classical, directly finite and co-hopfian while right quasi duo, right duo or commutative rings do not have necessarily these properties.

. The class of STRONGLY Ɵcl - continuous functions is considered in this paper. Studying about basic properties of STRONGLY Ɵcl - continuous functions, it is observed that these properties are similar to the properties of continuous functions. We denote by Scl(X)  the ring of all real valued STRONGLY Ɵcl - continuous functions on a topological space X . It is proved that if the range of a STRONGLY  Ɵcl- continuous function f  is a T1 - space, then f  is constant on quasi – components of its domain. Using this fact, we prove that for every topological space X , there is an ultra- Hausdorff  space Y  such that Scl(X) is isomorphic to C(Y).The behavior of these functions in relation to separation axioms is studied. We show that if f is a STRONGLY Ɵcl - continuous function from a topological space X  to a T0 - space Y , then X  is  ultra-Hausdorff. Topological properties of direct and inverse image of spaces with certain topological properties under STRONGLY Ɵcl - continuous functions are investigated. Among them, it is proved that the image of every cl-closure compact space under a STRONGLY Ɵcl - continuous function is compact. Finally the properties of the graphs of STRONGLY Ɵcl - continuous functions are discussed. It is proved that for every compact and Hausdorff space Y , strong Ɵcl - continuity of the function f from X to Y is equivalent to Ɵcl - closedness of  the graph of f relative to X .

The submodules with the property of the title (a submodule N of an R -module M is called STRONGLY dense in M, denoted by N£sd M, if for any index set I, ΠIN£d ΠIM) are introduced and fully investigated. It is shown that for each submoduleN of M there exists the smallest subsetD′ ÍM such that N+D′ is a STRONGLY dense submodule of M and D′ ∩ N=0. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module M, dense submodules in M are STRONGLY dense if and only if M £sd ~E (M), where ~E (M) is the rational hull of M. It is proved that R has no STRONGLY dense left ideal if and only if no nonzeroelement of every cyclic R -module M has a STRONGLY dense annihilator in R. Finally, some properties and new concepts related to strong density are studied.