Journal of Mathematical Extension
Volume:6 Issue: 3, Summer 2012

We study the structure of the set of nilpotent elementsin skew polynomial ring R[x; ], when R is an -Armendariz ring. We prove that if R is a nil -Armendariz ring and t = IR, then the set of nilpotent elements of R is an -compatible subrng of R. Also, it is shown that if R is an -Armendariz ring and t = IR, then R is nil-Armendariz. We give some examples of non -Armendariz rings which are nil -Armendariz. Moreover, we show that if t = IR for some positive integer t and R is a nil -Armendariz ring and nil(R[x][y; ]) = nil (R[x])[y], then R[x] is nil -Armendariz. Some results of [3] followas consequences of our results.

Let M and N be two non-zero right R-modules, M is called subhomomorphic to N in case there exist R−homomorphisms f: M ! N, g : N ! M such that gof is non-zero, and M is called strongly subhomomorphic to N in case there exist homomorphisms f :M! N, g : N ! M such that both fog and gof are non-zero. After establishing some basic properties of (strongly) subhomomorphic to a ring, it is shown that for a nonsingular ring R the class of injective right R -modules are subhomomorphic to R if and only if R is a semisimple ring.

This paper applies the nonparametric Data Envelopment Analysis (DEA) methodology to estimate and improve profit efficiency using an algorithm. The proposed method improves profit efficiency of inefficient units. This method uses the ratio of output prices to inputcosts in every decision making units. We have explained our method by an example.

Let R be a commutative ring with identity and M be a unitary R-module. A proper submodule N of M is 2− absorbing if r 1, r2, r3 2 R, m 2 M with r1r2r3m 2 M implies r1r2m 2 N or r1r3m 2 N or r2r3m 2 N. Let ' : S(M) −! S(M) [ {ø} be a function where S (M) is the set of all submodules of M. We call a proper submodule N of M a '-2-absorbing submodule if r1, r2, r3 2 R, m 2 M with r 1r2r3m 2 N −'(N) implies that r1r2m 2 N or r1r3m 2 N or r2r3m 2N. We want to extend 2-absorbing ideals to '-2-absorbing submodules and we show that-2-absorbing submodules enjoy analogs of many ofthe properties of 2-absorbing ideals.

We want to show the lineability of space of quasi-everywhere surjective functions, i.e. we want to show that the space contains a vector space with infinite dimension.

Data Envelopment Analysis (DEA), a popular linear programmingtechnique is useful to rate comparatively operational effiencyof decision Making Unit (DMU) based on the their deterministic inputoutputdata. The Malmquist productivity index in DEA, calculablewith the distance function, for measurement the productivity changeamong two variant time period or two variant group in the same time.This index is based on two factor of efficiency change index and atechnological change index. In this paper, we operate on the collectiveMalmquist productivity index, which performs clustering operationDMUs with classification into different levels of efficient frontier,and then we discuss on the relation between Malmquist index on theefficiency layers and their attractiveness and progress.

The Fermat-Weber location problem is to find a point in R n that minimizes the sum of the weighted Euclidean distances from mgiven points inRn. In this paper we consider the Fermat-Weber problemof one new facilitiy with respect ton unknown customers in order tominimizing the sum of transportation costs between this facility andthecustomers. We assumed that each customer is located in a nonemptyconvex closed bounded subset ofRn.

In this paper, a differential transform method (DTM) isused to find the numerical solution of the linear and nonlinear Westerveltequation. Exact solution can also be achieved by the known formsof the series solution. In this paper, we present the definition and operationof the three-dimensional differential transform and investigatethe particular exact solution of system of partial differential equationsthat usually arise in applied mechanic by a three-dimensional differentialtransform method. The numerical result of the present method ispresented and compared with the exact solution that is calculated bythe Laplace transform method.