module

In this paper we show that the category of measurable spaces is closed under coproducts in the category of sets. For an arbitrary ring R, we define measurable and probability right R-modules and we prove that the categories of these new objects are closed under kernels, cokernels and pushouts in the category of right R-modules. We also show that the category of measurable right R-modules is closed under coproducts and products in the category of right R-modules. We end this paper by giving some results about stochastically independence in the category of probability right R-modules.

For any R-module W, Dj(W) presented as the total of all J-small sub-modules. If A and B are sub-module of W, we say A is ⨁ Dj'supplement of B in W if W=A+B=A⨁A´, for A´↪−−W, and A⋂B≪jDj(A). If every sub-module has ⨁ Dj-supplemented, then W is ⨁ Dj-supplemented A sub-module A of W. If a sentence is conclusive, it is said to be cofinite i.e., WA is finitely generated. Also we introduce cofinite ⨁ Dj-supplemented if every cofinite sub-module of W has ⨁ Dj-supplemented.

Let R be a commutative ring with identity and let M be an R-module. This study presents the nearly locally hollow module that's a strong form of a hollow module. We present that an R-module M is nearly locally hollow if M has a unique semi-maximal sub-module that contains all small sub-modules of M. The current study deals with this class of modules and gives several fundamental properties related to this concept.

In this study‎, ‎$R$ and $M$ are assumed to be a commutative ring with non-zero identity $M$ and an $R$-module‎, ‎respectively‎. ‎Scalar Product Graph of $M$‎, ‎denoted by $G_R(M)$‎, ‎is a graph with the vertex-set $M$ and two different vertices $a$ and $b$ in $M$ are connected if and only if there exists $r$ belong to $R$ such that $a=rb$ or $b=ra$‎. ‎This paper studies some properties of such weakly perfect graphs‎.

Let R be a commutative ring with identity and M an R-module. The Scalar-Product Graph of M is defined as the graph GR(M) with the vertex set M and two distinct vertices x and y are adjacent if and only if there exist r or s belong to R such that x = ry or y = sx. In this paper , we discuss connectivity and planarity of these graphs and computing diameter and girth of GR(M). Also we show some of these graphs is weakly perfect.

ýIt is shown thatý ýif M is an Artinian module over a ringý ýRý, ýthen M has Noetherian dimension αý, ýwhere α is a countable ordinal numberý, ýif and only if ωα≤l(M)≤ωαý, ýwhere l(M) isý ýthe length of Mý, ýi.e., the least ordinal number such that the interval [0ý,ý l(M)) cannot be embedded in the lattice of all submodules of M.

In this paper, we find explicit solution to the operator equation TXS∗−SX∗T∗=A in the general setting of the adjointable operators between Hilbert C∗ modules, when T,S have closed ranges and S is a self adjoint operator.

Let M be an R-module and 0neqfinM∗=rmHom(M,R). We associate an undirected graph gf to M in which non-zero elements x and y of M are adjacent provided that xf(y)=0 or yf(x)=0. We observe that over a commutative ring R, gf is connected and diam(gf)leq3. Moreover, if Gamma(M) contains a cycle, then mboxgr(gf)leq4. Furthermore if |gf|geq1, then gf is finite if and only if M is finite. Also if gf=emptyset, then f is monomorphism (the converse is true if R is a domain). If M is either a free module with rmrank(M)geq2 or a non-finitely generated projective module there exists finM∗ with rmrad(gf)=1 and rmdiam(gf)leq2. We prove that for a domain R the chromatic number and the clique number of gf are equal.

Suppose T T and S S are Moore-Penrose invertible operators between Hilbert C*-module. Some necessary and sufficient conditions are given for the reverse order law (TS) dag =S dag T dag (TS)dag=SdagTdag to hold. In particular, we show that the equality holds if and only if Ran(T ∗ TS)subseteqRan(S) Ran(T∗TS)subseteqRan(S) and Ran(SS ∗ T ∗ )subseteqRan(T ∗ ), Ran(SS∗T∗)subseteqRan(T∗), which was studied first by Greville [{it SIAM Rev. 8 (1966) 518--521}] for matrices.

Let R be a commutative ring and M be an R-module with T(M) as subset, the set of torsion elements. The total graph of the module denoted by T(Γ(M)), is the (undirected) graph with all elements of M as vertices, and for distinct elements n,m∈M, the vertices n and m are adjacent if and only if n∈T(M). In this paper we study the domination number of T(Γ(M)) and investigate the necessary conditions for being Zn as module over Zm and we find the domination number of T(Γ(Zn)).

In his paper mentioned in the title, which appears in the same issue of this journal, MehdiRadjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminology of linear transformations. We add an additional translation of a ring-theoretic result to give a characterization of algebraicallyhyporeflexive transformations and the strict closure of the set of polynomials in a transformation T T.

We introduce variational inequality problems on Hilbert $C^*$-modules and we prove several existence results for variational inequalities defined on closed convex sets. Then relation between variational inequalities, $C^*$-valued metric projection and fixed point theory on Hilbert $C^*$-modules is studied.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The primary-like spectrum $Spec_L(M)$ is the collection of all primary-like submodules $Q$ such that $M/Q$ is a primeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ is a prime submodule for all $Qin Spec_L(M)$. This class contains the family of multiplication modules properly. The purpose of this paper is to introduce and investigate a new Zariski space of an RSP module, called Zariski-like space. In particular, we provide conditions under which the Zariski-like space of a multiplication module has a subtractive basis.

We consider the class M of R{modules where R is an associative ring. Let A be a module over a group ring RG, G be a group and let L(G) be the set of all proper subgroups of G. We suppose that if H 2 L(G) then A=CA(H) belongs to M. We study an RG{module A such that G 6= G0, CG(A) = 1. We consider the cases: 1) M is the class of all artinian {modules, R is either the ring of integers or the ring of p{adic integers; 2) M is the class of all nite R{modules, R is an associative ring; 3) M is the class of all nite R{modules, R= F is a nite eld.

In this paper we introduce the notions of G∗L-module and G∗L-module which are two proper generalizations of δ-lifting modules. We give some characteriza tions and properties of these modules. We show that a G∗L-module decomposes into 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.

In this paper, by considering the notion of extended BCK-module, we define the concepts of free extended BCK-module, free object in category of extended BCK-modules and we state and prove some related results. Specially, we define the notion of idempotent extended BCK-module and we get some important results in free extended BCK-modules. In particular, in category of idempotent extended BCK-modules, we give a method to make a free object on a nonempty set and in BCK-algebra of order 2, we give a method to make a basis for unitary extended BCK-modules. Finally, we define the notions of projective and productive modules and we investigate the relation between free modules and projective modules. In special case, we state the relation between free modules and productive modules.

Assume that $A$, $B$ are Banach algebras and that $m:Atimes Brightarrow B$, $m^prime:Atimes Arightarrow B$ are bounded bilinear mappings. We study the relationships between Arens regularity of $m$, $m^prime$ and the Banach algebras $A$, $B$. For a Banach $A $ -bimodule $B$, we show that $B$ factors with respect to $A$ if and only if $B^{**}$ is unital as an $A^{**} $ -module. Let $Z_{e^{primeprime}}(B^{**})=B^{**}$ where $e^{primeprime}$ is a mixed unit of $A^{**}$. Then $B^*$ factors on both sides with respect to $A$ if and only if $B^{**}$ has a unit as $A^{**} $ -module.

Let $R$ be a commutative ring. We write $mbox{Hom}(mu_A, nu_B)$ for the set of all fuzzy $R$-morphisms from $mu_A$ to $nu_B$, where $mu_A$ and $nu_B$ are two fuzzy $R$-modules. We make$mbox{Hom}(mu_A, nu_B)$ into fuzzy $R$-module by redefining a function $alpha:mbox{Hom}(mu_A, nu_B)longrightarrow [0,1]$. We study the properties of the functor $mbox{Hom}(mu_A,-):FRmbox{-Mod}rightarrow FRmbox{-Mod}$ and get some unexpected results. In addition, we prove that$mbox{Hom}(xi_p,-)$ is exact if and only if $xi_P$ is a fuzzy projective $R$-module, when $R$ is a commutative semiperfect ring.Finally, we investigate tensor product of two fuzzy $R$-modules and get some related properties. Also, we study the relationships between Hom functor and tensor functor.

The author studies the $bf R$$G$-module $A$ such that $bf R$ is an associative ring, a group $G$ has infinite section $p$-rank (or infinite 0-rank), $C_{G}(A)=1$, and for every proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank respectively) the quotient module $A/C_{A}(H)$ is a finite $bf R$-module. It is proved that if the group $G$ under consideration is locally soluble then $G$ is a soluble group and $A/C_{A}(G)$ is a finite $bf R$-module.

Let M be a module over a commutative ring R and let N be a proper submodule of M. The total graph of M over R with respect to N, denoted by T(ΓN(M)), have been introduced and studied in [2]. In this paper, A generalization of the total graph T(ΓN(M)), denoted by T(ΓN,I(M)) is presented, where I is an ideal of R. It is the graph with all elements of M as vertices, and for distinct m,n∈M, the vertices m and n are adjacent if and only if m∈M(N,I), where M(N,I)={m∈M:rm∈N for some r∈R−I}. The main purpose of this paper is to extend the definitions and properties given in [2] and [12] to a more general case.