Let r be a ring and n a submodule of a finitely generated faithful multiplication rmodule m. My writing project in ring theory started in 1983 after i taught a yearlong course in the. Kasch rings are named in honor of mathematician friedrich kasch. Duality of modules over topological rings 399 1 c is a cogenerator. Yousif in 14 introduced the notion of universally mininjective ring, a ring r is called right universally mininjective if s r \j 0.
M is the category consisting of all msubgenerated left r. 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. Aright koinjective ringis thereby right zip iff left kasch corollary 1. Modules are a generalization of the vector spaces of linear algebra in which the \scalars are allowed to be from an arbitrary ring, rather than a. 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. A right rmodule m is called right sninjective, where n is a right rmodule, if every rhomomorphism f k m. Quasidual module, kasch module, ikedanakayama 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.
If you are the owner for this file, please report abuse to 4shared. Rings, modules, and the total, by friedrich kasch and adolf mader. In what follows the term module will mean right module. Free, projective, and flat modules and ideals see also 19a citation kasch, friedrich. In section 2, we show that r is a right universally mininjective ring if and only if every simple right rmodule is ssinjective. We prove that if r is right artinian, or right fbn, or morita equivalent to a right duo ring, then every rmodule satisfies the condition.
In particular, we prove that a projective module m is rad. This book is a selfcontained elementary introduction to rings and modules, a topic constituting about half of a core course on algebra. Menini, linearly compact rings and strongly quasiinjective modules, rend. Introduction throughout this paper, ris an associative ring with identity, modules are right and unitary over it and s end rm is the ring of rendomorphisms of m. 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. On, semipotent and the total of rings and modules hakmi hamza department of mathematics, faculty of sciences, damascus university, syria. Many of theses contributions were presented at the international conference on ring and module theory held at hacettepe university in ankara, turkey, from. A characterization of artinian rings volume 30 issue 1 dinh van huynh, nguyen v. Now for any a2gwe have ea ayaa ayaa ae aas eis a right identity. For this, ais considered as module over the associative multiplication algebra ma and the category.
Throughout this note, ris a ring with an identity element and all modules are unital rmodules. As a consequence, ris a right kasch and a right pcsring if and only if ris a right qfring. Pdf we carry out a study of modules mr satisfying the property that every module in. Some characterization theorems of quasicoretractable modules in terms of kasch rings, cs modules, monocoretractable modules, projective modules, epiretractable modules and retractable modules will be. Julius zelmanowitz mathematics publications endomorphism rings of torsionless modules article journal of algebra 5 1967, 325341. Ris a right kasch ring if and only if the injective hull of every simple right rmodule is neat. The proofs are treated with full details retaining the classroom flavour. Kasch originally called artinian rings whose proper ideals have. 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. Regularity and substructures of hom friedrich kasch springer.
Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. Rings over which every simple module has an injective cover. Recent trends in rings and algebras, murcia, 37 june 20 rings over which every simple module has an injective cover y. 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. Locally injective modules and locally projective modules. Introduction in this paper we introduce and investigate the notion of sinjective modules and rings. 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. A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not. 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. 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. 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.
Also torsion modules over a topological ring and graded modules over a graded ring form categories of the type. In this paper we provide various properties of rad. 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. Following 1, a module rm is said to be kasch provided that every simple module in.
Model structures on modules over dingchen rings gillespie, james, homology, homotopy and applications, 2010. The concept of left kasch rings were generalized to modules in paper 1. The goal of the present paper is to study left and right toto modules and toto rings. If gis a group of even order, prove that it has an element a6esatisfying a2 e. 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 section 3, some classical results on modules and rings that satisfy chain conditions over submodules are extended, and sometimes improved. Best reference books ring theory and rings and modules. As a consequence, we will also deduce the main results of 11, 12. We prove that if r is right artinian, or right fbn, or morita equivalent to a right duo ring. W e are able to characterize fully kasch modules m r.
Knapp, basic algebra, digital second edition east setauket, ny. A shorter proof of goldies theorem article canadian math. 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. Several sufficient conditions for a module to be fully kasch are given which are also necessary in case the module satisfies a property. C of the set of complex numbers containing the numbers 0. Friedrich kasch, regularity and substructures of hom english isbn. I plan to cover much of the material in the first three parts pages 1118. A ring r is called a right pvring if every simple rmodule is pinjective. Hyperrigid operator systems and hilbert modules shankar, p. Morita, duality for modules and its applications to the theory of rings with minimum conditions, sci. Modules and rings london mathematical society monographs.
Textbook writing must be one of the cruelest of selfinflicted tortures. For example, if the base ring r is semisimple, then all r modules would automatically be semisimple. Transferring this notion to modules we call a left rmodule a kasch module if it contains a copy of every simple module in. Injective modules, selfinjective rings see also 16l60 16d40. In this paper, we will discuss and study the structure of quasicoretractable modules.
Let r be a commutative ring with identity and m an rmodule. More recently zelmanowitz 16 has called a module regular if every finitely generated submodule is a projective. Analogously the notion of a left kasch ring is defined, and the two properties are independent of each other. Regularity and substructures of hom friedrich kasch. Quasicoretractable modules asianeuropean journal of. Regularity and substructures of hom books pics download. An associative ring r is a left kasch ring if it contains a copy of every simple left rmodule. When the context is clear, we shall write em for erm. Fully kasch modules and rings mohammad baziar academia. Kasch modules have studied by series of authors, see 1, 6, 5. Fuller, rings and categories of modules, springer gtm 1992.
The aim of this paper is to characterize and investigate this class of modules. Received 18042010 accepted 27092010 abstract let m and n be two modules over a ring r. It is proved that if a kasch module m is finitely generated and quasipinjective, then there is a bijective. Lr and sr modules, another generalization of cohopfian and hopfian modules. M is the category consisting of all msubgenerated left rmodules. We obtain a characterization of right kasch rings as follows. The entire material including exercise is fully class tested. Good tilting modules and recollements of derived module categories, ii chen, hongxing and xi, changchang, journal of the mathematical society of japan, 2019. Let r be a right kasch ring such that each cyclic submodule of the injective envelope err embeds in a free module. The following graduate level textbooks are recommended for further study. Semisimple rings are of particular interest to algebraists. Furthermore, every simple left rmodule is isomorphic to a minimal left ideal of r, that is, r is a left kasch ring. For example, if the base ring r is semisimple, then all rmodules would automatically be semisimple. Generalized fitting modules and rings sciencedirect.
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