Modules in which every surjective endomorphism has a δ-small kernel

In this paper,we introduce the notion of δ-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of δ-Hopfian modules by proving that a ring R is semisimple if and only if every R-module is δ-Hopfian. Also, we show that for a ring R, δ(R) = J(R) if and only if for all R-modules, the conditions δ-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that δ-Hopfian property is a Morita invariant. Further, the δ-Hopficity of modules over truncated polynomial and triangular matrix rings are considered

African Index Medicus: Improving access to African health information

Information flow is the key to improving health development, especially in the developing countries. African medical publications are poorly represented in the major medical electronic databases. African Index Medicus is a joint initiative between WHO and AHILA to store regionally-generated biomedical information. Proposed in 1980 and initiated in 1993, AIM was reactivated in 2005 and now emphasizes full text accessibility and web publishing. To promote use of AIM and health knowledge-sharing, the WHO has provided national focal points with training, computers and scanners. Publishing still faces challenges of strengthening networks of national focal points and African medical editors as well as transferring technology and experience to African countries. There still remain the more basic constraints of costs, training, marketing and low status of both research and publishing. The Special Programme for Research and Training in Tropical Diseases further found problems of underfunding, irregular publication schedules, low quality articles and lack of international visibility. A TDR survey in early 2006 revealed that there is increased health research and journal activities in African countries; however, there are still challenges of quality, content and accessibility. Since its inception in 2002 the Forum of African Medical Editors has held three training workshops for editors to correct some of these problems. AIM will soon be part of the WHO Global Health Library; both provide access to health information which will contribute to meeting the Millennium Development Goals for health. These initiatives promise more health information for resource-poor settings, especially in Africa. South African Family Practice Vol. 49 (2) 2007: pp. 5-1

L-zero-divisor graphs of direct products of L-commutative rings

L-zero-divisor graphs of L-commutative rings have been introduced and studied in [5]. Here we consider L-zero-divisor graphs of a finite direct product of L-commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the L-ziro-divisor graph of a L-ring when extending to a finite direct product of L-commutative rings

On L-ideal-based L-zero-divisor graphs

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied

An ideal-based zero-divisor graph of direct products of commutative rings

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings

Some remarks on Prüfer modules

We provide several characterizations and investigate properties of Prüfer modules. In fact, we study the connections of such modules with their endomorphism rings. We also prove that for any Prüfer module M, the forcing linearity number of M, fln(M), belongs to {0,1}

A graph associated to proper non-small ideals of a commutative ring

summary:In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G(R)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter, girth, clique number, cut vertex, planar property and independence number. Further, it is shown that the independence number of a small graph of a ring $R$ is equal to the number of its maximal ideals and the domination number of small graph is at most 2

A co-ideal based identity-summand graph of a commutative semiring

summary:Let $I$ be a strong co-ideal of a commutative semiring $R$ with identity. Let $\Gamma_{I} (R)$ be a graph with the set of vertices $S_{I} (R) = \{x \in R\setminus I: x + y \in I$ for some $y \in R \setminus I\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \in I$. We look at the diameter and girth of this graph. Also we discuss when $\Gamma_{I} (R)$ is bipartite. Moreover, studies are done on the planarity, clique, and chromatic number of this graph. Examples illustrating the results are presented