Algebras, Rings and Modules Non-commutative Algebras and Rings © 2016 by Taylor & Francis Group, LLC © 2016 by Taylor & Francis Group, LLC Algebras, Rings and Modules Non-commutative Algebras and Rings Michiel Hazewinkel Dept. of Pure and Applied Mathematics Centrum Wiskunde & Informatica Amsterdam, Netherlands and Nadiya Gubareni Institute of Mathematics Częstochowa University of Technology Częstochowa, Poland p, A SCIENCE PUBLISHERS BOOK © 2016 by Taylor & Francis Group, LLC GL--Prelims with new title page.indd ii 4/25/2012 9:52:40 AM GL--Prelims with new title page.indd ii 4/25/2012 9:52:40 AM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20151214 International Standard Book Number-13: 978-1-4822-4505-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2016 by Taylor & Francis Group, LLC PREFACE The theory of rings and modules is one of the most fundamental domains of modern algebra. This volume is a continuation of the first two volumes of Algebra, Rings and Modules by M. Hazewinkel, N. Gubareni and V.V. Kirichenko. Volume 3, published by the American Mathematical Society, is about Lie algebras and Hopf algebras and largely independent of the other volumes. It systematizes and presents the main results of the structure theory of some special classes of non-commutative rings. The book presents both the basic classical theory and more recent results related to current research such as the structure theory of some special classes of rings, which arise in many applications. Some of the topics covered include quivers, partially ordered sets and their representations, as well as such special rings as hereditary and semihereditary rings, serial rings, semidistributive rings and modules over them. Some results of this book are new and have, so far, been published in journals only. All results are given with complete proofs which are based on the material contained in the book. We assume that the reader is familiar with the basic concepts of theory of rings and modules. Nevertheless for the reader’s convenience Chapter 1 summarizes the basic ring-theoretic notions and results considered in our previous books. Proofs of all results presented can be found in these books via corresponding citations which have been inserted in parentheses. In mathematics, specifically in the area of abstract algebra, it is often interesting to construct new objects using objects already known. In group theory, ring theory, Lie algebra theory there are a variety of different constructions such as crossed, skew, smash products which are very important as sources of various counter examples. Some of the main ring constructions, such as a finite direct product of rings, group rings, matrix rings, path algebras and graded algebras were considered in our previous books. Chapter 2 represents the definitions and some properties of some of these basic general constructions of rings and modules, such as direct product, semidirect product, direct sum, crossed product of rings, polynomial and skew polynomial rings, power and skew power series rings, Laurent polynomial rings and Laurent power series rings, generalized matrix rings, formal triangular matrix rings, and G-graded rings. The theory of valuation rings was first related only with commutative fields. Discrete valuation domains are, excepting only fields, the simplest class of rings. Nevertheless they play an important role in algebra and algebraic geometry. But there is also a noncommutative side of this theory. In the noncommutative case there are different generalizations of valuation rings. The first generalization, © 2016 by Taylor & Francis Group, LLC vi Algebras, Rings and Modules valuation rings of division rings, was obtained by Schilling in [283], who introduced the class of invariant valuation rings and systematically studied them in [284]. Another significant contribution in non-commutative valuation rings was made by N.I. Dubrovin who introduced a more general concept of a valuation ring for simple Artinian rings and proved a number of significant nontrivial properties about them [73], [71], [72]. These rings are named Dubrovin valuation rings after him. Dubrovin valuation rings found a large number of applications. In Chapter 3 most of the basic results for valuation rings and discrete valuation rings are described and discussed briefly. Section 3.1 is devoted to valuation rings of fields. In this section we give the main properties of these rings and describe different definitions of them. Section 3.2 presents the basic results about discrete valuation domains. We give the structure of these rings and a number of equivalent definitions. In Section 3.3 we describe noncommutative invariant valuation rings of division rings, and the main properties of these rings and their equivalent definitions are described. Some examples of noncommutative non-discretely-valued valuation rings are given in Section 3.4. The main properties and structure of noncommutative discrete valuation rings are presented in Section 3.5. In Section 3.6 we briefly discuss total valuation rings which are more general than invariant valuation rings. Section 3.7 is devoted to other types of valuation rings with zero divisors. We consider some valuation rings of commutative rings with zero divisors and the Dubrovin valuation rings. Finally, in Section 3.8 we consider the approximation theorems for special kinds of noncommutative valuation rings, in particular for locally invariant rings and Dubrovin valuation rings. We also give some corollaries from these theorems for noncommutative discrete valuation rings. Chapter 3 may be considered as a short introduction to the theory of valuation rings. More information about valuation rings of division rings can be found in [284], and more about Dubrovin valuation rings, semihereditary and Prüfer orders in simple Artinian rings can be found in the book [233]. The concepts of homological dimensions of rings and modules were discussed in [146, Chapter 6] and [147, Chapter 4]. Chapter 4 considers some other questions connected with these notions. For the reader’s convenience Section 4.1 summarizes the basic concepts and results on projective and injective dimensions of modules, and global dimensions of rings. The concepts and results on flat dimensions of modules and weak dimensions of rings are presented in Section 4.2. In Section 4.3 we present various examples of rings with different global dimensions. The homological characterization of some classes of rings, such as semisimple, right Noetherian, right hereditary, right semihereditary, semiperfect, right perfect and quasi-Frobenius rings are considered in Section 4.4. Duality over Noetherian rings, which is given by the covariant functor * = Hom A (−, A) was considered in [147, Section 4.10]. For an arbitrary ring A this functor induces a duality between the full subcategories of finitely generated projective right A-modules and left A-modules. The main properties of this functor and torsionless modules, and the relationship between them are studied in Section 4.5. The basic properties of flat modules were considered in [146, Section 5.4], and in [147, Section 5.1]. Further properties of flat modules are studied in Section 4.6. In © 2016 by Taylor & Francis Group, LLC Preface vii particular, the main theorem of this section, which was proved by S. Chase [41], gives equivalent conditions for a direct product of any family of flat modules to be flat. As the corollary of this theorem we obtain homological characterization of semihereditary rings, proved by S. Chase in [41]. Section 4.7 gives necessary and sufficient conditions under which a formal triangular matrix ring is right (left) hereditary. Chapter 5 contains a short introduction to the theory of uniform, Goldie and Krull dimensions of rings and modules. Uniform modules and their main properties are considered in Section 5.1, where the uniform dimension of modules is also introduced and studied. Modules of finite Goldie dimension, a notion due to A. Goldie, are considered in Section 5.2. The notions of singular and nonsingular modules are introduced in Section 5.3, where there are also studied the main properties of such modules and some properties of nonsingular rings. In Section 5.4 the results of this theory are applied to prove a theorem which gives equivalent conditions for a ring being a Goldie ring. This theorem includes the famous Goldie theorem which was proved in [100], [101], [102] (see also [146, Section 9.3]). This well-known theorem gives necessary and sufficient conditions for a ring to have a semisimple classical quotient ring. In 1966 L. Small generalized this theorem and described Noetherian rings which have Artinian classical rings of quotients [292]. A variant of Small’s theorem without the Noetherian hypothesis was obtained by R.B. Warfield, Jr. [321]. In Section 5.5 the proofs of these results are given. The notion of the classical Krull dimension as a powerful tool for arbitrary commutative Noetherian rings was considered by W. Krull. There are a few different generalizations of this concept for the case of noncommutative rings. One of them, the classical Krull dimension introduced by G. Krause in [197], is considered in Section 5.6. The more important generalization, the concept of Krull dimension is considered in Section 5.7. The module-theoretic form of this notion in the general case for noncommutative rings was introduced by R. Rentschler and P. Gabriel in 1967, [270]. Note that not all modules have Krull dimension, but each Noetherian module has Krull dimension. The Krull dimension of any Artinian module is equal to 0. So in some sense the Krull dimension of a module can be considered as a measure which shows of how far the module is from being Artinian. The basic properties of Krull dimension are studied in this section. In Section 5.8 we consider some relationships between the concepts of classical Krull dimension and Krull dimension. An important role in the theory of rings and modules is played by various finiteness conditions. Many types of finiteness conditions on rings can be formulated in terms of d.c.c. (descending chain condition) or a.c.c. (ascending chain condition) on suitable classes of one-sided ideals. The d.c.c. (minimal condition) on right (resp., left) ideals defines right Artinian (resp., left Artinian) rings. Analogously, right (resp., left) Noetherian rings are defined as rings which satisfy the maximal condition, or the a.c.c. on right (resp. left) ideals. These rings were considered in [146, Chapter 3]. Section 6.1 gives some examples of Noetherian rings connected with the basic constructions of rings considered in Chapter 2. Section 6.2 considers various finiteness conditions for rings and modules, and relations between them. Dedekind-finite rings, orthogonally finite rings, stably finite rings, unit-regular rings and IBN rings are examples of rings which are considered in this section. © 2016 by Taylor & Francis Group, LLC viii Algebras, Rings and Modules FDI-rings, i.e., rings with a finite decomposition of the identity into a sum of pairwise orthogonal primitive idempotents, form the next class of rings with finiteness conditions. These rings are considered in Section 6.3. In Section 6.4 we prove the main theorem which gives a criterium for a semiprime FDI-ring to be decomposable into a direct product of prime rings. Chapter 7 is devoted to the important problems connected with uniqueness of decompositions of modules into direct sums of indecomposable modules. The famous Krull-Remak-Schmidt theorem was already considered in [146], where in Section 10.4 this theorem was proved for the case of finite direct sums of modules with local endomorphism rings. Actually, G. Azumaya proved this theorem in [12] for infinite direct summands in the general case for Abelian categories with some additional condition. In this chapter the proof of this theorem is given for the case of infinite direct sums of modules with local endomorphism rings following to Peter Crawley and Bjarni Jo´nsson. They proved this theorem using the exchange property, which was introduced in 1964 for general algebras, [61]. From that time this notion has become an important theoretical tool for studying rings and modules. Some properties of modules having the exchange property are studied in Section 7.1. It is proved that the 2-exchange property is equivalent to the finite exchange property for arbitrary modules, and that the 2-exchange property is equivalent to the exchange property for indecomposable modules. These results were obtained by P. Crawley and B. Jo´nsson for general algebras in [61] and R.B. Warfied, Jr. for Abelian categories in [321]. In this section we also prove the important result obtained by R.B. Warfied, Jr. in [317], which states that an indecomposable module has the exchange property if and only if its endomorphism ring is local. The proof of the Azumaya theorem for infinite direct sums of modules is given in Section 7.2. The cancellation property notion and some properties of modules having the cancellation property are considered in Section 7.3. At the end of this chapter we consider different classes of rings connected with exchange rings. Section 7.4 is devoted to the study of properties and some structural theorems for exchange rings. Generally speaking, the class of semisimple rings has been studied most extensively. Their structure is completely described by the Wedderburn-Artin theorem. Semisimple rings are also very simple from the point of view of homological properties of modules over them. These are rings whose global homological dimension is equal to zero. Hereditary rings immediately follow semisimple ones in terms of a homological classification. According to theorem 4.3.8, r.gl.dim A < 1 if and only if A is a right hereditary ring. The structure of hereditary rings is not so well studied as in the case of semisimple rings. Chapter 8 is devoted to the study of the structure and main properties of hereditary rings. In addition, semihereditary rings, which are close to hereditary rings, are considered in this chapter. In Section 4.5 it was shown that for a large class of rings (coherent, semiperfect, right serial, right Noetherian) being right semihereditary implies being left semihereditary. In Section 8.1 this result is proved for orthogonally finite rings. The main results of Chapters 4 and 5 are applied to study the properties of right hereditary and right semihereditary rings in Section 8.2. In particular the Goldie theorem it is proved there. This theorem gives equivalent conditions for a domain A © 2016 by Taylor & Francis Group, LLC Preface ix to be right Ore. There is also the important Small theorem which states that a right Noetherian right hereditary ring is a right order in a right Artinian ring. The structure and properties of some classes of right hereditary (semihereditary) prime rings are considered in Section 8.3. In particular we prove an important theorem which states the relationship between Dubrovin valuation rings, semihereditary orders and Bézout orders in simple Artinian rings. We also consider right Noetherian hereditary (semihereditary) semiperfect prime rings. The next generalization, following hereditary and semihereditary rings, are piecewise domains. These rings were first introduced and studied by R. Gordon and L.W. Small in 1972. Section 8.4 considers properties of piecewise domains and their relationships with hereditary and semihereditary rings. It is proved that a piecewise domain is a nonsingular ring. This section also gives a proof of the theorem which states that a right perfect piecewise domain is semiprimary. This theorem first was proved by M. Teply in 1991. The notion of a triangular ring was first introduced by S.U. Chase in 1961 for semiprimary rings. In 1966 L. Small extended this notion to Noetherian rings and proved that a right Noetherian right hereditary ring is triangular. M. Harada in 1964 introduced the notion of generalized triangular rings and proved that any hereditary semiprimary ring is isomorphic to a generalized triangular ring with simple Artinian blocks along the main diagonal. In 1980 Yu.A. Drozd extended the notion of a triangular ring to FDI-rings and described the structure of right hereditary (semihereditary) FDI-rings. Section 8.5 introduces the notions of triangular and primely triangular rings which includes all notions of a triangular ring mentioned above. The main result of this section gives the structure of piecewise domains in terms of primely triangular rings. This theorem was proved by R. Gordon and L.W. Small in 1972 and it states that any piecewise domain is a primely triangular ring. From this statement there easily follows the theorem obtained by L.W. Small in 1966 about the structure of right Noetherian right hereditary rings. Section 8.6 gives the criterion for a triangular FDI-ring to be right hereditary or right semihereditary, which was obtained by Yu.A. Drozd in 1980. In Section 8.7 the results of Section 8.6 are applied to different concrete classes of rings. In particular, we give the criterion for a right Noetherian primely triangular ring to be right hereditary. From this result there follows the famous decomposition theorem of Chatters which states that a Noetherian hereditary ring is a direct sum of rings each of which is either an Artinian hereditary ring or a prime Noetherian hereditary ring. Section 8.8 is devoted to the study of hereditary species and tensor algebras, which were introduced by Yu.A. Drozd. Chapter 9 is devoted to the further study of serial rings which were considered in [146, Sections 12, 13]. In this chapter we present the structure theorems for various different classes of serial nonsingular rings. In Section 9.1 we consider serial right Noetherian piecewise domains. We prove that for a serial right Noetherian ring being a piecewise domain is equivalent to being a right hereditary ring. Section 9.2 is devoted to the study of the structure of serial nonsingular rings. In particular, it is given the main result of R.B. Warfield, Jr. who proved that for a right serial ring being right semihereditary is equivalent to be © 2016 by Taylor & Francis Group, LLC x Algebras, Rings and Modules right nonsingular. We also give another equivalent conditions for right serial right semihereditary rings. In this section we study the structure of serial nonsingular rings and show that any serial nonsingular ring has a classical ring of quotients which is an Artinian ring. Section 9.3 is devoted to serial rings with Noetherian diagonal. We consider the prime quiver of such rings and describe the structure of serial nonsingular rings with Noetherian diagonal. The Krull intersection theorem is very important and well known for Noetherian commutative rings. In Section 9.4 we consider some versions of this theorem for noncommutative rings. Section 9.5 is devoted to the problems connected with the Jacobson conjecture which states that for any Noetherian ring with Jacobson radical R the intersection ∩Rn = 0. This conjecture is true for any commutative Noetherian ring, but it is still n≥0 open for noncommutative Noetherian rings in general. There are various classes of noncommutative Noetherian rings for which the Jacobson conjecture holds. In particular, the Jacobson conjecture holds for discrete valuation rings which is shown in Section 3.5. A.V. Jategaonkar [167] and G. Cauchon [39] have shown that Jacobson’s conjecture hold for fully bounded Noetherian rings. T.H. Lenagan [216] have proved that this conjecture also holds for Noetherian rings with right Krull dimension one. In this section we prove that the Jacobson conjecture holds for Noetherian SPSD-rings and Noetherian serial rings. The book is written on a level accessible to advanced students who have some experience with modern algebra. It will be useful for those new to the subject as well for researchers and serves as a reference volume. © 2016 by Taylor & Francis Group, LLC