Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. First, we study multiplication $R$-modules $M$ where $R$ is a one dimensional Noetherian ring or $M$ is a finitely generated $R$-module. In fact, it is proved that if $M$ is a multiplication $R$-module over a one dimensional Noetherian ring $R$, then $Mcong I$ for some invertible ideal $I$ of $R$ or $M$ is cyclic. Also, a multiplication $R$-module $M$ is finitely generated if and only if $M$ contains a finitely generated submodule $N$ such that $Ann_R(N)= Ann_R(M)$. A submodule $N$ of $M$ is called dense in $M$, if $M=sum_varphivarphi(N)$ where $varphi$ runs over all the $R$-homomorphisms from $N$ into $M$ and $R$-module $M$ is called a weak $pi$-module if every non-zero finitely generated submodule is dense in $M$. It is shown that a faithful multiplication module over an integral domain $R$ is a weak $pi$-module if and only if it is a Prufer prime module.
For a group $G$, $pi_e(G)$ and $s_m(G)$ are denoted the set of orders of elements and the number of elements of order $m$ in $G$, respectively. Let ${rm nse}(G)={s_m(G) | min pi_e(G)}$. An arbitrary finite group $M$ is NSE characterization if, for every group $G$, the equality ${rm nse}(G)={rm nse}(M)$ implies that $Gcong M$. In this paper, we are going to show that the non-Abelian finite simple groups $A_9$, $A_{10}$, $A_{12}$, $U_4(3)$, $U_5(2)$, $U_6(2)$, $S_6(2)$, $O_8^+(2)$ and $HS$ are characterizable by NSE.
In this paper, some categorical properties of the category { Pre-Dcpo} of all pre-dcpos; pre-ordered sets which are also pre-directed complete, with pre-continuous maps between them is considered. In particular, we characterize products and coproducts in this category. Furthermore, we show that this category is neither complete nor cocomplete. Also, epimorphisms and monomorphisms in {Pre-Dcpo} are described.Finally, some adjoint relations between the category {Pre-Dcpo} and others are considered.More precisely, we consider the forgetful functors between this category and some well-known categories, and study the existence of their left and right adjoints.
A finite group $G$ is called textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = left .$ In 29, Moori posed the question of finding all the $(p,q,r)$ triples, where $p, q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is a $(p,q,r)$-generated. In this paper we establish all the $(p,q,r)$-generations of the symplectic group $Sp(6,2).$ GAP 20 and the Atlas of finite group representations 33 are used in our computations.
This paper studies homoderivations satisfying certain conditions on semigroup ideals of near-rings. In addition, we include some examples of the necessity of the hypotheses used in our results.
In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$ in which $amin N$ with ${rm Ann}_M(a)=(0)$ implies that $min N$ for each $ain R$ and $min M$). We show that every simple submodule, direct summand, divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${rm Ann}_M(a)not=(0)$ or $aM=M$, for every $ain R$) and strongly $uz$-module (i.e., an $R$-module $M$ such that $aMsubseteq a^2M$, for every $ain R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian $R$-module is a $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then $M$ is a $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$ is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.
Throughout this paper, all rings are commutative with identity and all modules are unital. Let $R$ be a ring and $M$ be an $R$-module. Then $M$ is called a multiplication module provided for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N=IM$. Also $M$ is said to be a comultiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $ N=(0:_MI)$. In this paper, we introduce the notions of reduction and coreduction of submodules, integral dependence, integral codependence, integral closure and $Delta$-closure over multiplication and comultiplication modules.
In this paper, we extend the concept of small subhypermodules to alltypes of hypermodules and give nontrivial examples for this concept. As an application, we define and study lifting hypermodules via small subhypermodules.
This paper, by considering the notion of a state residuated lattice morphism in the class of state residuated lattices, investigates some classical theorems namely the going up and lying over theorems. Results show that each state residuated lattice morphism fulfills these theorems. Also, some properties about prime filters of residuated lattices are obtained which are given in the paper.
A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the notions of (perfect-total) 2-dominating set and (perfect-total) 2-domination numbers on vague graphs are introduced and some properties are investigated. Especially, it is proven that in any strong vague graph on a Petersen graph, any minimal 2-dominating set is a minimal perfect 2-dominating set and minimal dominating set. Then, the concepts of (total) 2-cobondage set and (total) 2-cobondage number in vague graphs are expressed and related results obtained. Finally, an application related to Fire Stations and Emergency Medical centers is provided.
This paper introduces a novel concept of Boolean function--based hypergraph with respect to any given T.B.T(total binary truth table). This study defines a notation of kernel set on switching functions and proves that every T.B.T corresponds to a Minimum Boolean expression via kernel set and presents some conditions on T.B.T to obtain a Minimum irreducible Boolean expression from switching functions. Finally, we present an algorithm and so Python programming(with complete and original codes) such that for any given T.B.T, introduces a Minimum irreducible switching expression.
In this paper, we introduce a notion of ultra central approximate identity for Banach algebras which is a generalization of the bounded approximate identity and the central approximate identity. Using this concept we study pseudo-contractibility of some matrix algebras among $ell^1$-Munn algebras. As an application, for the Brandt semigroup $S=M^{0}(G,I)$ over a non-empty set $I$, we show that $ell^{1}(S)$ has an ultra central approximate identity if and only if $I$ is finite. Also we show that the notion of pseudo-contractibility and contractibility are the same on $ell^{1}(S)^{**}$, where $S$ is the Brandt semigroup.
Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$. Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the paper.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity 1 and $M$ a graded $R$-module. A proper graded submodule $C$ of $M$ is called a graded classical prime submodule if whenever $r,sin h(R)$ and $min h(M)$ with $rsmin C$, then either $rmin C$ or $smin C$. In this paper, we introduce the concept of graded $J_{gr}$-classical prime submodule as a new generalization of graded classical submodule and we give some results concerning such graded modules. We say that a proper graded submodule $N$ of $M$ is textit{a graded }$J_{gr}$textit{-classical prime submodule of }$M$ if whenever $rsmin N$ where $r,sin h(R)$ and $min h(M)$, then either $rmin N+J_{gr}(M)$ or $smin N+J_{gr}(M)$, where $J_{gr}(M)$ is the graded Jacobson radical.
Let M be a unitary left R-module where R is a ring with identity. The co-intersection graph of proper submodules of M, denoted by Omega(M), is an undirected simple graph whose the vertex set V (Omega) is a set of all non-trivial submodules of M and there is an edge between two distinct vertices N and K if and only if N + K not equal M. In this paper we investigate connections between the graph-theoretic properties of Omega(M) and some algebraic properties of modules. We characterize all of modules for which the co-intersection graph of submodules is connected. Also the diameter and the girth of Omega(M) are determined. We study the clique number and the chromatic number of Omega(M).
Suppose that G is a groupoid with binary operation ⊗. The pair (G,⊗) is said to be a gyrogroup if the operation ⊗ has a left identity, each element a ∈ G has a left inverse and the gyroassociative law and the left loop property are satisfied in G. In this paper, a method for constructing new gyrogroups from old ones is presented and the structure of subgyrogroups of these gyrogroups are also given. As a consequence of this work, five 2-gyrogroups of order 2^n, n≥ 3, are presented. Some open questions are also proposed.
In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct generating sets of $G$ if $\mathrm{Cay}(G,A)$ has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.
Let $M$ be a module over a ring $R$. We call $M$,$delta$-$H$-supplemented provided for every submodule $N$ of $M$ there is a direct summand $D$ of $M$ such that $M=N+X$ if and only if $M=D+X$ for every submodule $X$ of $M$ with $M/X$ singular. We prove that $M$ is $delta$-$H$-supplemented if and only if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $(N+D)/Nll_{delta} M/N$ and $(N+D)/Dll_{delta} M/D$.
Let $ mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ mathbb {Z}_{n} $ is defined as the graph $ G( mathbb {Z}_{n} ) $ with the vertex set $ mathbb {Z}_{n} $ and two distinct vertices $a,b$ are adjacent if and only if $a-bin Uleft( mathbb {Z}_{n}right)$, where $ Uleft( mathbb {Z}_{n}right) $ is the set of units of $ mathbb {Z}_{n} $. Let $Gamma ( mathbb {Z}_{n} ) $ be the complement of $ G( mathbb {Z}_{n} ) $. In this paper, we determine the independence number of $ Gamma ( mathbb {Z}_{n} ) $. Also it is proved that $ Gamma ( mathbb {Z}_{n} ) $ is well-covered. Among other things, we provide condition under which $ Gamma ( mathbb {Z}_{n} ) $ is vertex decomposable.
In this paper, it is shown that $ (mathcal{V}, mathfrak{X}) $ is a Schur pair if and only if the Baer-invariant of an $mathfrak{X}$-group with respect to $ mathcal{V}$ is an $mathfrak{X}$-group. Also, it is proved that a locally $mathfrak{X}$ class inherited the Schur pair property of , whenever $mathfrak{X}$ is closed with respect to forming subgroup, images and extensions of its members. Subsequently, many interesting predicates about some generalizations of Schur's theorem and Schur multiplier of groups will be concluded.
