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.

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.

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.

A simple connected graph G of order n  3 is a Strongly 2-multiplicative if there is an injective mapping f: V (G)! f1; 2; : : :; ng such that the induced mapping h: A! Z + de ned by h(P) = Q3 i=1 f(vj i ), where j1; j2; j3 2 f1; 2; : : :; ng, and P is the path homotopy class of paths having the vertex set fvj1; vj2; vj3 g, is injective. Let  (n) be the number of distinct path homotopy classes in a Strongly 2-multiplicative graph of order n. In this paper we obtain an upper bound and also a lower bound for  (n). Also we prove that triangular ladder, P2 J Cn, Pm J Pn, the graph obtained by duplication of an arbitrary edge by a new vertex in path Pn and the graph obtained by duplicating all vertices by new edges in a path Pn are Strongly 2-multiplicative.

This paper investigates the thermodynamic properties of Strongly interacting gravitational systems. By deriving the partition function for such systems, the authors obtain expressions for various thermodynamic quantities, including internal energy, specific heat, Helmholtz free energy, entropy, chemical potential, and pressure. The paper introduces an interaction parameter that quantifies the degree of non-ideality in the system, and explores its effects on the thermodynamic properties. The authors also calculate the moments generating function, which provides information about the distribution of particle positions. Additionally, the distribution function for the system is derived. The paper highlights the existence of an upper bound temperature beyond which the partition function becomes negative, indicating a limit on the validity of the model.