Extended Modules and Ore Extensions

In this paper,we study extended modules for a special class of Oreextensions.We will assume that R is a ring and A will denote the Ore extensionA:=R[X1,...,xn;σ]for whichσis an automorphism of R,xixj=xjxi and xir=σ(r)xi,for every 1<i,j≤n.With some extra conditions over the ring R,wewill prove Vaserstein's,Quillen's patching,Horrocks',and Quillen-Suslin's theoremsfor this type of non-commutative rings.

Principally Quasi-Baer Modules

In this paper,we give the equivalent characterizations of principally quasi-Baer modules,and show that any direct summand of a principally quasi-Baer module inherits the property and any finite direct sum of mutually subisomorphic principally quasi-Baer modules is also principally quasi-Baer.Moreover,we prove that left principally quasi-Baer rings have Morita invariant property.Connections between Richart modules and principally quasi-Baer modules are investigated.

Fundamental algorithm and computational codes for the light beam propagation in high power laser system

The fundamental algorithm of light beam propagation in high powerlaser system is investigated and the corresponding computational codes are given. It is shown that the number of modulation ring due to the diffraction is related to the size of the pinhole in spatial filter (in terms of the times of diffraction limitation, i.e. TDL) and the Fresnel number of the laser system; for the complex laser system with multi-spatial filters and free space, the system can be investigated by the reciprocal rule of operators.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect.The class of coperfect modules lies properly bet ween the class of locally artinian modules and the class of semiartinian modules.Let R be a commutative ring with identity.We show that every semiartinian Ti-module is coperfect if and only if R is a T-ring.It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if m/m^(2)is a finitely genera ted R-module for every maximal ideal m of R.

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤

Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Let R = ⊙n〉0 Rn be a standard graded ring, a ∩ ⊙n〉0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i 〉 0, the n-th graded component Hiα（M, N）n of the i-th generalized local cohomology module of M and N with respect to a vanishes for all n 〉〉 0. Some sufficient conditions are pro- posed to satisfy the equality sup{end（Hiα （M, N）） [ i _〉 0} = sup{end（HiR＋ （M, N）） ｜ i 〉 0}. Also, some sufficient conditions are proposed for the tameness of Hiα（M, N） such that i = fRα＋（M,N） or i = cdα（M,g）, where fRα＋（M,N） and cdα（M,g） denote the R＋- finiteness dimension and the cohomological dimension of M and N with respect to a, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of Hjα（M, N）, where j is the first or last non-minimax level of Hiα（M, N）.