Forcing Linearity Numbers for Modules over PID's

Let V be a module over a principal ideal domain. Then V = M N where M is divisible and N has no nonzero divisible submodules. In this paper we determine the forcing linearity number for V when N is a direct sum of cyclic modules. As a consequence, the forcing linearity numbers of several classes of Abelian groups are obtained.     

Forcing Linearity Numbers for Modules over Simple Domains

If R is a simple Noetherian ring and V an R-module, then every homogeneous function on V is an endomorphism, i.e., the forcing linearity number, fln(V), of V is zero, unless R is a domain. Here we consider the problem of finding forcing linearity numbers for modules over simple Noetherian domains. As an application we find the forcing linearity numbers for all finitely generated modules over the first Weyl algebra.     

Forcing Linearity Numbers of Semisimple Modules over Integral Domains

In this paper we determine the forcing linearity numbers for semisimple modules over integral domains.     

Some Remarks on Multiplication and Projective Modules II

All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6 Ali , M. M. , Smith D. J. ( 2004 ). Some remarks on multiplication and projective modules . Communications in Algebra 32 : 3897 – 3909 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ? Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.     

Idempotent and Nilpotent Submodules of Multiplication Modules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.     

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N?=?IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M)?=?{PM?|?P?∈?Spec(R) and P???M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

Forcing matchings on square grids

Let G be a graph that admits a perfect matching. The forcing number of a perfect matching M of G is defined as the smallest number of edges in a subset S M, such that S is in no other perfect matching. We show that for the 2n × 2n square grid, the forcing number of any perfect matching is bounded below by n and above by n². Both bounds are sharp. We also establish a connection between the forcing problem and the minimum feedback set problem. Finally, we present some conjectures about forcing numbers in other graphs.     

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.     

L-模糊自然数的乘法运算和幂运算

在L是完全分配格时,定义了L-模糊自然数的乘法运算和幂运算,研究了乘法运算、幂运算的交换律、结合律以及乘法对加法的分配律等性质。     

乘法运算和虚数单位的几何定义

对于自然数,乘法是加法一种简明的表达式;但由自然数系扩展为整数系时,乘法却需要补充的几何定义,以加深对运算律的理解。为此,任何有向线段a与(-1)的乘积定义为有向线段a绕其起点逆时针旋转π角所生成的有向线段;任何有向线段a与j的乘积定义为有向线段a绕其起点逆时针旋转π/2角所生成的有向线段,由此可推导出j即是虚数单位,j=i=(-1)～(1/2)。eiθ既是单位向量,又是平面向量的乘法旋转算子。文中还阐明了复数的指数形式为平面向量的最佳表达式,以及平面向量三种乘法的对应关系。     

Multiplication Modules and a Theorem of P. F. Smith

P. F. Smith [7, Theorem 8] gave sufficient conditions on a finite set of modules for their sum and intersection to be multiplication modules. We give sufficient conditions on an arbitrary set of multiplication modules for the intersection to be a multiplication module. We generalize Smith"s theorem, and we prove conditions on sums and intersections of sets of modules sufficient for them to be multiplication modules. This revised version was published online in June 2006 with corrections to the Cover Date.     

Forcing axioms via ground model interpretations

We study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that ${\sigma }^g$ satisfies φ. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are:

• •Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to:

  • A principle for rank 1 names (equivalently, nice names) for subsets of ${\omega }_1$.

  • A principle for rank 2 names for sets of reals.
• •λ-bounded forcing axioms are equivalent to name principles. Bagaria's characterisation of $\mathsf{BFA}$ via generic absoluteness is a corollary.

We further systematically study name principles where φ is a notion of largeness for subsets of ${\omega }_1$ (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms.     

Monomial Modules and Endo-Monomial Modules

We determine the multiplicity algebras and multiplicity modules of a p-monomial module. For a general p-group P, we find a sufficient and necessary condition for an endo-monomial P-module to be an endo-permutation P-module, and prove that a capped indecomposable endo-monomial P-module is of p ^′-rank. At last, we give an alternative definition of the generalized Dade P-group.     

Weak Multiplication Modules

In this paper we characterize weak multiplication modules.     

$FI$-$t$-提升模和$t$-quasi-dual Baer模

本文引入$FI$-$t$-提升模和$t$-quasi-dual Baer模的概念并给出两者的联系.证明富足补模$M$为$FI$-$t$-提升模当且仅当$M$的每个完全不变$t$-coclosed子模为$M$的直和项当且仅当$\bar{Z}^{2}(M)$为$M$的直和项且$\bar{Z}^{2}(M)$为$FI$-$t$-提升模当且仅当$M$同时为$t$-quasi-dual Baer 模和$FI$-$t$-$\mathcal{K}$-模.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.