Goodearl 14 gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings 27, 31, 15. Received 18042010 accepted 27092010 abstract let m and n be two modules over a ring r. Model structures on modules over dingchen rings gillespie, james, homology, homotopy and applications, 2010. Modules and rings london mathematical society monographs. It is proved that if a kasch module m is finitely generated and quasip. Rings, modules, and the total, by friedrich kasch and adolf mader. If you are the owner for this file, please report abuse to 4shared. Transferring this notion to modules we call a left rmodule a kasch module if it contains a copy of every simple module in. Mares 9 has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. A right rmodule m is called pinjective if every homomorphism from a principal right ideal of r to m can be given by a left.
This book is a selfcontained elementary introduction to rings and modules, a topic constituting about half of a core course on algebra. Kasch modules have studied by series of authors, see 1, 6, 5. Also torsion modules over a topological ring and graded modules over a graded ring form categories of the type. Hyperrigid operator systems and hilbert modules shankar, p.
Furthermore, every simple left rmodule is isomorphic to a minimal left ideal of r, that is, r is a left kasch ring. On the other hand, kasch and mader, kam04 study lr and sr modules, another generalization of cohop. Preface vii notes to the reader xvii partial list of notations xix partial list of abbreviations xxiii 1 free modules, projective, and injective modules 1 1. C of the set of complex numbers containing the numbers 0. Good tilting modules and recollements of derived module categories, ii chen, hongxing and xi, changchang, journal of the mathematical society of japan, 2019. For this, ais considered as module over the associative multiplication algebra ma and the category.
Semisimple rings are of particular interest to algebraists. Friedrich kasch, regularity and substructures of hom english isbn. Best reference books ring theory and rings and modules. Fully kasch modules and rings mohammad baziar academia. When the context is clear, we shall write em for erm. My writing project in ring theory started in 1983 after i taught a yearlong course in the.
The entire material including exercise is fully class tested. In section 3, some classical results on modules and rings that satisfy chain conditions over submodules are extended, and sometimes improved. Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. We prove that if r is right artinian, or right fbn, or morita equivalent to a right duo ring. A ring r is called a right pvring if every simple rmodule is pinjective. The aim of this paper is to characterize and investigate this class of modules. M is the category consisting of all msubgenerated left r. Regularity and substructures of hom friedrich kasch. Using results from the monograph of mohamed and mueller on discrete modules, it is shown that the endomorphism ring of a discrete module, modulo its radical, is a direct product of full linear rings. The goal of the present paper is to study left and right toto modules and toto rings. A characterization of artinian rings volume 30 issue 1 dinh van huynh, nguyen v. Applying these notions to rings, as modules over themselves, we obtain interesting classes of rings, some of them are well known and very interesting, as the strongly. Exercises and solutions in groups rings and fields 5 that yaayat ethen yaae e hence yaa e.
It is proved that if a kasch module m is finitely generated and quasipinjective, then there is a bijective. For example, if the base ring r is semisimple, then all r modules would automatically be semisimple. Pdf we carry out a study of modules mr satisfying the property that every module in. Fuller, rings and categories of modules, springer gtm 1992. As a consequence, we will also deduce the main results of 11, 12. As a consequence, ris a right kasch and a right pcsring if and only if ris a right qfring. The proofs are treated with full details retaining the classroom flavour. Knapp, basic algebra, digital second edition east setauket, ny. Julius zelmanowitz mathematics publications endomorphism rings of torsionless modules article journal of algebra 5 1967, 325341. I plan to cover much of the material in the first three parts pages 1118. Generalized fitting modules and rings sciencedirect. In ring theory, a subfield of abstract algebra, a right kasch ring is a ring r for which every simple right r module is isomorphic to a right ideal of r.
Regularity and substructures of hom friedrich kasch springer. Morita, duality for modules and its applications to the theory of rings with minimum conditions, sci. Throughout this note, ris a ring with an identity element and all modules are unital rmodules. Analogously the notion of a left kasch ring is defined, and the two properties are independent of each other kasch rings are named in honor of mathematician friedrich kasch. A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not. Lr and sr modules, another generalization of cohopfian and hopfian modules. Let r be a ring and n a submodule of a finitely generated faithful multiplication rmodule m. Injective modules, selfinjective rings see also 16l60 16d40. In section 2, we show that r is a right universally mininjective ring if and only if every simple right rmodule is ssinjective. Recent trends in rings and algebras, murcia, 37 june 20 rings over which every simple module has an injective cover y.
Some characterization theorems of quasicoretractable modules in terms of kasch rings, cs modules, monocoretractable modules, projective modules, epiretractable modules and retractable modules will be given. Recall that a ring r is called right kasch if every simple right module embeds in r. Menini, linearly compact rings and strongly quasiinjective modules, rend. Now for any a2gwe have ea ayaa ayaa ae aas eis a right identity. Kasch originally called artinian rings whose proper ideals have. Rings over which every simple module has an injective cover. Rings, modules, and the total kasch and mader developed the theory of the total over the last 1020 years and put it together for the first time in this book. M is the category consisting of all msubgenerated left rmodules. Yousif in 14 introduced the notion of universally mininjective ring, a ring r is called right universally mininjective if s r \j 0. Quasicoretractable modules asianeuropean journal of. The following graduate level textbooks are recommended for further study. R is a right fully kasch ring if and only if the module r r is fully kasch, in whic h case r is a left perfect ring.
In this paper we provide various properties of rad. A shorter proof of goldies theorem article canadian math. In what follows the term module will mean right module. Let r be a right kasch ring such that each cyclic submodule of the injective envelope err embeds in a free module. Zhu recall that a ring r is called left kasch 7 if every simple leftrmodule embeds in rr,equivalently, r rt 0 for every maximal left ideal t of r. Analogously the notion of a left kasch ring is defined, and the two properties are independent of each other. Many of theses contributions were presented at the international conference on ring and module theory held at hacettepe university in ankara, turkey, from. Textbook writing must be one of the cruelest of selfinflicted tortures. Locally injective modules and locally projective modules. If gis a group of even order, prove that it has an element a6esatisfying a2 e. Regularity and substructures of hom books pics download. Several sufficient conditions for a module to be fully kasch are given which are also necessary in case the module satisfies a property. More recently zelmanowitz 16 has called a module regular if every finitely generated submodule is a projective. Quasidual module, kasch module, ikedanakayama module.
For example, if the base ring r is semisimple, then all rmodules would automatically be semisimple. W e are able to characterize fully kasch modules m r. Aright koinjective ringis thereby right zip iff left kasch corollary 1. Kasch 1 introduction given a ring r with unity, we denote by modi. We obtain a characterization of right kasch rings as follows. Let r be a commutative ring with identity and m an rmodule. Some characterization theorems of quasicoretractable modules in terms of kasch rings, cs modules, monocoretractable modules, projective modules, epiretractable modules and retractable modules will be. People who are searching for free downloads of books and free pdf copies of these books introduction to rings and modules by c musili, introductory lectures on rings and modules london mathematical society student texts by john a beachy, rings and categories of modules graduate texts in mathematics by frank w anderson and kent r fuller, module theory. In this paper, we will discuss and study the structure of quasicoretractable modules. An associative ring r is a left kasch ring if it contains a copy of every simple left rmodule. Introduction in this paper we introduce and investigate the notion of sinjective modules and rings. Duality of modules over topological rings 399 1 c is a cogenerator. Ris a right kasch ring if and only if the injective hull of every simple right rmodule is neat.
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