Ring Endomorphisms with Large Images

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism S of a prime one-sided noetherian ring R is injective whenever the image S (R) contains an essential left ideal L of R. If additionally S(L) = L, then S is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.Comment: To appear in Glassgow Mthematical Journal, 12 page

Decomposition of Singular Matrices into Idempotents

In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over B\'ezout domains as to when such a matrix is a product of idempotent matrices.Comment: 15 page

ADS modules

We study the class of ADS rings and modules introduced by Fuchs. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that R is ADS as a right R-module must be either right self-injective or indecomposable as a right R-module. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of completely ADS modules and characterize completely ADS semiperfect right modules as direct sum of semisimple and local modules.Comment: 7 page

Editor’s note

This special issue contains papers written by participants to the conference "Noncommutative rings and their Applications" that was held in Lens (France) at the Science faculty of the Universit\'e d'Artois. The meeting gave the experts from different domains the opportunity to exchange their views, share their research, and learn from one another new results and problems in a friendly atmosphere. They were 55 researchers, graduate and postdoctoral students from USA, Canada, Poland, Czech Republic, Italy, Spain, Germany, Portugal, Egypt, Senegal, New Zeland, South Africa, Algeria, Turkey, Mexico, Brazil, Uruguay, Indonesia...and France! The interplay between ring and coding theory was emphasized by the very nice and interesting course "Linear codes from the axiomatic point of view" given by Jay Wood. The four invited speakers: Fr\'ed\'erique Oggier, Christophe Reutenauer, Angel del Rio and Irfan Siap contributed greatly to the success of this conference. The topics of the Jay Wood's course and the 39 talks presented at the conference are well represented by this special issue of Jacodesmath. They cover pure ring theory such as nil *-clean rings, radical classes or algebras such as Grassman algebras and Steenrod algebras and many papers relating these subjects with coding theory such as MacWilliams extension theorem, dual codes, Lee and Hamming weight. So this volume will be particularly useful for mathematicians at the confluent of these two branches. This meeting was supported by the Laboratoire de Math\'ematiques de Lens (LML), by different bodies from the Universit\'e d'Artois (RI, BQR), as well as by a regional organization (the Fédération des Laboratoires de Math´ematiques du Nord pas de Calais). We would like to thank all the participants for their efforts and enthusiasm. A very warm thanks to the colleagues who kindly agreed to referee the papers. Their expertise, promptitude, and professionalism improved the quality of the articles in this volume. Many thanks are due to the editorial staff of the Jacodesmath journal for the very efficient way they managed the process of preparing and publishing these proceedings

A description of quasi-duo Z-graded rings

A paraître dans "Communications in Algebra"A description of right (left) quasi-duo Z-graded rings is given. It shows, in particular, that a strongly Z-graded ring is left quasi-duo if and only if it is right quasi-duo. This gives a partial answer to a problem posed by Dugas and Lam in [1]

Noncommutative polynomial maps

Accepté pour publication dans "Journal of Algebra and its applications"; 16 pages.Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a fi nite fi eld F_q[t;S ], where S is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring F_q[x]