An R-module A is called GF-regular if, for each a ∈ A and r ∈ R, there exist t ∈ R and a positive integer n such that r(n)tr(n)a = r(n)a. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by M GF(A). Furthermore, the radical properties of A are investigated; we proved that if A is an R-module and K is a submodule of A, then MGF(K) = K∩M GF(A...

Recent measurements of cosmic-ray electron and positron fluxes by PAMELA and ATIC experiments may indicate the existence of annihilating dark matter with large annihilation cross section. We discuss its possible relation to other astrophysical/cosmological observations : gamma-rays, neutrinos, and big-bang nucleosynthesis. It is shown that they give stringent constraints on some annihilating da...

Let G be a digraph with n vertices and let A(G) be its adjacency matrix. A monic polynomial f (x) of degree at most n is called an annihilating polynomial of G if f (A(G)) = 0. G is said to be annihilatingly unique if it possesses a unique annihilating polynomial. Difans and diwheels are two classes of annihilatingly unique digraphs. In this paper, it is shown that the complete product of difan...

Top-down design methodology is one of the widely used approaches to the design of complex concurrent systems. In this approach, the speciication of a system is decomposed into a set of submodules whose concurrent behavior is equivalent to that of the system speciication. The following problem is of particular importance when using this methodology: given the speciication of a system and some of...

Starting from a Caldeira-Leggett model for the interaction of a system with an environment, Joichi, Matsumoto, and Yoshimura have reconsidered the derivation of the quantum Boltzmann equation. They find an extra term that accounts for the effects of virtual particles, and which drastically changes the results for relic densities of stable, weakly interacting massive particles (WIMPs), and for t...

In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely generated. (2) Even if a submodule of a finitely generated module is finitely generated, the minim...

in this paper we introduce the notions of g∗l-module and g∗l-module whichare two proper generalizations of δ-lifting modules. we give some characteriza tions and properties of these modules. we show that a g∗l-module decomposesinto a semisimple submodule m1 and a submodule m2 of m such that every non-zero submodule of m2 contains a non-zero δ-cosingular submodule.

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

During this search all rings are commutative and modules unitary. In we introduced the concept of Restrict Nearly semi-prime Sub-modules as generalization give some basic properties, examples charactarizations concepts stablished sufficient conditions on to be

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...