DIAGONAL REDUCTION OF MATRICES OVER RINGS

Transcription

1 Bohdan Zabavsky DIAGONAL REDUCTION OF MATRICES OVER RINGS Mathematical Studies Monograph Series VOLUME XVI VNTL Publishers

2 , Lviv University, Ukraine, Lviv University, Ukraine Reviewers:, Institute of Mathematics, NAS of Ukraine, Kyiv University, Ukraine, Kyiv University, Ukraine Recommended by the Council of the Department of Mechanics and Mathematics, Lviv University, Ukraine (????) Editor: Michael Zarichnyi c 2012 VNTL Publishers All rights reserved. No part of this publication may be reproduced or used in any form by any means without the permission of the publisher. VNTL Publishers: P.O. Box 10249, Lviv 79006, Ukraine info@vntl.com Prepared from L A TEX file Printed in Ukraine ISBN

3 Contents PREFACE 5 1 PRELIMINARIES 18 2 FINITE STABLE RANGE RINGS n-hermite rings Compact space of minimal prime ideals Diagonalization of matrices Direct finiteness of matrix rings Cases of Hermite quotient ring Stable range 1 rings BEZOUT RINGS Maximal nonprincipal ideals Adequate rings and their generalizations Maximal and prime specter of commutative Bezout rings without zero divisors ELEMENTARY DIVISOR RINGS Simple elementary divisor rings Block-diagonal reduction of matrices Elementary matrix reduction Matrix reduction over locally countable commutative Bezout rings

4 4 CONTENTS 4.5 Matrix reduction over Bezout ring with unique maximal nonprincipal ideal, that satisfies Dubrovin condition Weak diagonal reduction of matrices over von Neumann regular rings Elementary divisor rings with L condition Matrix reduction over almost atomic Bezout rings without zero divisors of stable range 1, that satisfies Dubrovin condition Matrix reduction over commutative Bezout rings without zero divisors of stable range 1 in localizations Simultaneous reduction of pair

5 This book is dedicated to the memory of my mother Mariya PREFACE The problem of diagonalization of matrices is a classic one. Its prototype is the Gauss theorem on the equivalence of arbitrary matrix over the field to a diagonal matrix with ones and zeros on the main diagonal. The first results of this type concerning the integers were obtained in 1861 by H. Smith [218]. He proved that, by elementary transformations of rows and columns, every matrix with integer elements can be reduced to a diagonal form, and each diagonal element is a divisor of the following one (therefore, the diagonal form of the matrix with the condition of divisibility of the diagonal elements is often called the Smith form). Later, the Smith theorem was extended to different classes of rings, which, generally speaking, do not coincide. Thus, Dixon [133], Wedderburn [238], van der Waerden [230], and Jacobson [166] extended this theorem over different classes of commutative and noncommutative Euclidean rings and commutative principal ideal rings without zero divisors and Teichmüller [224] received a complete solution for the noncommutative principal ideal rings without zero divi- 5

6 6 Preface sors (and, in a different formulation, Asano [93]). These results in the most complete form are set out in Jacobson s monograph [13]. These results led to the introduction the notion of an elementary divisor ring by Kaplansky [172]. Recall that a matrix over an associative ring has a canonical diagonal reduction if it can be reduced to a diagonal form by left and right multiplication by some unimodular matrices of the corresponding sizes and so that each diagonal element is a full divisor of the following one. If each matrix over the ring has canonical diagonal reduction, then such a ring is called an elementary divisor ring [172]. In the same paper, Kaplansky showed that over an elementary divisor ring arbitrary finitely represented module can be decomposed into a direct sum of cyclic modules. In the case of commutative rings the reverse statement is proved, namely: if finitely represented module over a ring can be decomposed into a direct sum of cyclic modules, then this ring is an elementary divisor ring [181]. This result is a partial solution to the problem of Warfield [234]: over which rings every finitely represented module can be decomposed into a direct sum of cyclic submodules. Thus, for the commutative rings, the Warfield problem is equivalent to the problem of description of the elementary divisor rings. In the noncommutative case, the problem is not fully resolved. A solution of this problem for the class of generalized uniserial rings is obtained by Drozd [14]. Note that Kirichenko [67] dealt with close concepts. Other characteristics of rings with the above properties are given by Kaplansky [172] and Lafon [177], however, as Feith [82] writes: Whether they can be considered as an answer to this question depends on the tastes of the reader. An important role in studying of the elementary divisor rings is played by the Hermitian rings. A ring is called right (left) Hermitian if all 1 2 (2 1) matrix have diagonal re-

7 PREFACE 7 duction over this ring. An Hermitian ring is a ring which is both right and left Hermitian. The property of ring to be Hermitian plays an important role in considering the possibility of diagonal reduction of matrices, since this condition is present in all known necessary and sufficient conditions of theorems on elementary divisor rings. At the same time, under an Hermitian ring one often understands a ring over which an arbitrary stably free module is free [201]. Note that any Hermitian ring is a finitely generated principal ideal ring, i.e. a ring in which an arbitrary finitely generated left or right ideal is principal [172] (these rings are generalizations of the principal ideal rings). Therefore the problem of studying finitely generated principal ideal rings is actual. The main examples of finitely generated principal ideal rings are the ring of continuous real-valued functions on completely regular Hausdorff space [144], the ring of polynomials (power series) over the field of rational numbers with integer free term [73, 158], the ring of all integer algebraic numbers [177], the ring of analytic functions in the complex plane [156]. Since the class of commutative finitely generated principal ideal rings without zero divisors is axiomatizable, it is ultraclosed [121]. This fact allows us using ultraproducts to construct new examples of commutative finitely generated principal ideal rings. Kazimirsky [61], Cohn [129] and Amitsur [85] independently showed that if a finitely generated right principal ideal ring has no zero divisors, the the intersection of arbitrary two principal right ideals is a principal right ideal. Therefore, any finitely generated right principal ideal ring without zero divisors is a right Bezout ring. This concepts was introduced in 1963 by Cohn [129]. The commutative Bezout rings without zero divisors were also studied by Jaffard and Bourbaki [167], [4]. The name of the rings expresses the fact that any two coprime

8 8 Preface elements satisfy the Bezout identity au bv = 1. In recent years, under the Bezout rings one typically understands the rings of both right and left finitely generated ideals. The Bezout rings and their properties are studied in the publications [4, 8, 22, 25, 61, 62, 73, 77, 81, 84, 85, 95, 96, 97, 98, 99, 100, 102, 103, 104, 107, 108, 110, 121, 122, 123, 124, 125, 126, 127, 128, 129, 134, 139, 144, 146, 168, 169, 170, 171, 189, 198, 206, 211, 212, 213, 215, 227, 236, 241]. Since any elementary divisor ring is an Hermitian, it is also a Bezout ring. The question arises whether an arbitrary Bezout ring is an elementary divisor ring? Gilman and Henriksen [144] constructed an example of a commutative Bezout ring which is not an elementary divisor ring. Moreover, therein an example is constructed of a commutative Hermitian ring which is not an elementary divisor ring. Consequently, the classes of Bezout rings, Hermitian rings and elementary divisor rings do not coincide. So the following problem arises naturally: under what conditions any Bezout ring is an Hermitian ring? Note that, as Menal, Monkazi [191] showed, a regular ring is an elementary divisor ring if and only if it is an Hermitian ring. A similar result for the semigroup rings is obtained by Chounard [118], and the for adequate commutative rings by Levy Shores, and Larsen [181]. This means that, for these classes of rings, the answer to the question whether this Bezout ring is Hermitian is simultaneously the answer to the questions whether this ring is an elementary divisor ring. Note that, as Amitsur [85] showed, if a Bezout ring has no zero divisors, then this ring is Hermitian. Moreover, if all zero divisors of a Bezout ring lie in the Jacobson radical, then this ring is also Hermitian [172]. Henriksen posed the question: Is any semilocal commutative Bezout ring Hermitian [158]? In 1974, in [181] it is shown that

9 PREFACE 9 any commutative Bezout ring with finite number of minimal prime ideals is Hermitian. And here, as a consequence, an affirmative answer to the question of Henriksen is received. In another direction, the rings with compact space of minimal prime ideals were actively studied [92, 135, 154, 163, 182, 190, 203], and therefore Henriksen formulated the questioned whether any Bezout ring with compact space of minimal prime ideals is Hermitian [217]. Let us emphasize the important role of Bezout rings with compact space of minimal prime ideals in solving the known Kaplansky open problem of description of rings over which an arbitrary finitely generated module decomposes into a direct sum of cyclic ones [102, 103, 104, 138, 217, 228]. In the same paper [158] Henriksen, while studying the question of closeness of the elementary divisor rings with respect to homomorphic images poses the question: if R is a commutative Bezout ring such that R/J(R) is Hermitian, is then R Hermitian as well? Investigating unitally regular rings, Henriksen [159] showed that any matrix over this ring is equivalent to a diagonal one. In the same paper he posed the question on describing regular elementary divisor rings (in the sense of Henriksen), i.e. the rings over which an arbitrary matrix is equivalent to a diagonal matrix. Menal, Monkazi cite165 have shown that over a regular ring an arbitrary matrix is equivalent to a diagonal matrix if and only if a ring is an Hermitian ring. At the same time, it turned out that there are whole classes of regular rings, for example, the class of separative regular rings over which a square matrix is equivalent to a diagonal matrix [87]. Note that this fact holds also for the semi-chain rings [185]. Later it turned out that there are regular rings over which the matrices of only certain sizes are equivalent to the diagonal ones [116]. Therefore the question naturally raises of reduction to the diagonal form of not all matrices but matrices of only certain

10 10 Preface form or of a certain size. Let us make a little clarification concerning various definitions that can be found in the issues of diagonalization of matrices. Recall the definition proposed by Kaplansky [172]. Definition A ring R is called an elementary divisor ring, if, for an arbitrary matrix A of order n m over R, there exist unimodular matrices P GL n (R) and Q GL m (R) such that 1) P AQ = D is a diagonal matrix, D = (d i ); 2) Rd i+1 R Rd i d i R. Definition A ring R is called an ID-ring [164], if every idempotent matrix over R is diagonalizable, i.e., for every matrix E of order n over R such that E 2 = E, there exist unimodular matrices P, Q GL n (R) such that 1) P EQ = D is a diagonal matrix; 2) D 2 = D = D t. Definition A ring R is called an idempotently diagonalizable [229], if for every square matrix A of order n over R there exist unimodular matrices P, Q GL n (R) such that 1) P AQ = D is a diagonal matrix; 2) D 2 = D. Definition A ring R is called an diagonalizable, if for every square matrix A of order n over R there exist unimodular matrices P GL n (R) and Q GL m (R) such that P AQ = D is a diagonal matrix [159, 191]. These rings are often called Henriksen elementary divisor rings [159, 11].

11 PREFACE 11 In many publications the issue of diagonalization of arbitrary regular matrix is considered (see [11, 87, 116, 165]). Recall that a matrix A over a ring R is called it regular if over R there exists a nonzero matrix X such that AXA = A. In [87] the question of diagonalization of regular matrices over exchange rings is discussed. Note that these classes of rings are closely related, but they are different. For example, Hilbruk and Van Hil [164] showed that over any ID-domain and over any Artinian ring every regular matrix can be idempotently diagonalized. Note that there are different definitions of equivalence of matrices. Definition We say that matrices A and B (not necessarily of the same size) are it Steinitz equivalent [220] if there exist matrices P 1, P 2, Q 1, Q 2 (not necessarily unimodular) of the corresponding size over R such that B = P 1 AQ 1 and A = P 2 BQ 2. Definition We say that matrices A and B (again of not necessarily identical size) are it Krull equivalent / [175] (denoted A K B), if there are unimodular matrices P and Q, and blocks of zero matrices of appropriate sizes such that ( ) B 0 = P 0 0 ( ) A 0 Q. 0 0 Definition We say that matrices A and B over a ring R are equivalent if there are unimodular matrices P and Q of appropriate size such that B = PAQ.

12 12 Preface This definition is considered a classic one [172]. The above definition played an important role in connection with the following problem. Let R be an associative ring with identity. Find necessary and sufficient conditions in order that a homogeneous system of linear equations n a ij x j = b j, j=1 i = 1, 2,..., m, a ij, b i R be consistent in R? [61, 175, 220] Noether and Hentselt solved it for the case of polynomial rings of one unknown. Sneper generalized their results to the case of rings of polynomials in n unknowns over a field, and then to the case of commutative ring with unit without zero divisors, in which the condition of cutting of ascending chains of ideals holds [61]. Steinits [220] and Krul1 [175] obtained a solution to the problem for the case of the ring of integer algebraic numbers. Kazimirsky solved this problem for the case of a main right (left) ideal ring [61]. In order to generalize these results to the case of Bezout rings one has to investigate elementary divisor rings both commutative and noncommutative comprehensively. Most of the known classes of elementary divisor rings significantly depend on the conditions of termination of ascending chains of ideals. The first example of a classical elementary divisor ring without conditions of termination of ascending chains of ideals was found by Vanderbern in already in 1915, namely, such is the ring of analytic functions [239]. In a more abstract form, this example allowed Helmer [156] to introduce a new class of elementary divisor rings called the class of adequate rings. The adequate rings with zero divisors in the Jacobson radical were considered by Kaplansky [172]. Gillman and Henriksen

13 PREFACE 13 showed that any commutative regular ring is adequate [145]. At the same time, the structure of these rings remains little studied. Among the famous results, we note the following one: each nonzero prime ideal of an adequate ring is contained in a unique maximal ideal; any self-local ring is adequate if and only if it is either intersection of a finite number of pairwise independent valuation rings without zero divisors with common field of fractions or finite direct sum of valuation rings [181]. In the paper [105] an example is provided of a commutative not adequate Bezout ring without zero divisors in which any nonzero prime ideal is contained in a unique maximal ideal. The first example of a non-adequate Bezout ring without zero divisor which is an elementary divisor ring is found by Henriksen [158]. Note that the adequate rings without Bezout condition were investigated in [196, 223]. Questions concerning closeness of the classes of elementary divisor rings and adequate rings with respect to distribution of algebraic constructions were also investigated by some authors. For example, Dubrovin proved closeness of the class of commutative elementary divisor rings with respect to transition to projective limits of projective systems of certain type [16]. Some authors imposed restrictions either on cardinality of the ring or cardinality of its maximum spectrum. Thus, in the momograph [173] it is formulated, and the paper [106] it is proved that any commutative Bezout ring without zero divisors whose set of maximal ideals is at most countable must be an elementary divisor ring. In this paper [106], it is also shows that any commutative Bezout ring of Krull dimension 1 is adequate. Also, the equivalence is established of the problem of elementary divisors for the Bezout rings without zero divisors and the problem of pole accessability for linear systems over such rings. In the paper [181], it is proved that any commutative Bezout ring in which each element belongs to only a finite

14 14 Preface set of maximal ideals is an elementary divisor ring. Note the results of [138, 139, 157, 160, 162, 163, 187, 188], where the spectrum of Bezout rings was investigated. Note that in [139] an example of a commutative Bezout ring without zero divisors R in which there is the main ideal I such that the space of minimal prime ideals of R/I is not compact. This example provides a negative answer to the following open question: let R be a reduced coherent ring and let I be a finitely generated ideal of R; is the space of minimal prime ideals of the ring R/I compact? Shores and Wiegand proved that every commutative Bezout ring of Noetherian spectrum is an elementary divisor ring [217]. The unimodular matrices are most comfortable in applied problems, and just when they can be represented as a finite product of elementary matrices. This holds, for example, over a field, over the ring of integers or, more generally, over any Euclidean ring. But in general, not every unimodular matrix can be generated by elementary matrices [101, 130]. Therefore, the problem of studying such rings is actual. Such studies were initiated by Cohn [123] and continued by Bergman. These studies have shown that there are rings that generalize Euclidean rings and such that arbitrary matrix over these rings can be reduced to canonical diagonal form by elementary transformations of rows and columns. By [172], this implies that, over these rings, arbitrary unimodular matrix can be obtained from the identity matrix by finite sequence of elementary transformation of rows and columns. Kuke [130] and Bohi [101] investigated which classes of entire rings of algebraic numbers are the described rings. Romanov [77] showed that the left main ω-euclidean ring without zero divisors is a ring on which an arbitrary matrix can be reduced to the canonical diagonal form (with certain restrictions on the elementary divisors) by elementary transformations of rows and column.

15 PREFACE 15 In connection with these studies, we note the following fact. In the algebraic K-theory, the notion of K 1 -functor that assigns to a certain ring its Whitehead group is prevalent. In the commutative case, the Whitehead group can be decomposed into a direct sum of units of the ring and the factor-group of a special linear group by some subgroup generated by elementary matrices. It is clear that over the discussed above rings the Whitehead group is isomorphic to the group of ring units [94]. The noncommutative elementary divisor rings were studied quite fragmentarily. In addition to the mentioned results concerning regular rings [159] and principal ideal rings [13], one should emphasize the results of Cohn [125], which showed that the right general Bezout ring without zero divisors is an elementary divisor ring with certain conditions on the diagonal elements. In [127], some examples of such rings are constructed. Dubrovin [17] showed that any semilocal and semiprimary Bezout ring R is an elementary divisor ring if and only if for any element a inr there exists b R such that RaR = br = Rb. In the paper [53] studied quasi-duo-elementary divisor domains; this allowed Tuhanbaev to show that a distributive ring is an elementary divisor ring if and only if it is a duo-ring (see [80]). Note that the duo-ring of discrete valuation is used by Kirichenko and Hibina [68] when describing structure generalozed one-row and semimaximal rings. The elementary divisor rings and their properties are also studied in [6, 7, 9, 12, 16, 17, 24, 26, 27, 30, 62, 63, 65, 70, 71, 72, 73, 74, 75, 76, 77, 80, 118, 125, 138, 143, 145, 150, 151, 174, 176, 202, 211, 212, 216, 217, 223, 225, 240]. There is, as we see, a lot of diagonal matrices equivalent to a given matrix. To discuss the question of uniqueness define

16 16 Preface the cokernel of a matrix A of order m n over a ring R as a right R-module coker A = R m /AR n. Obviously, if matrices A and B are equivalent, then cokera = cokerb. Note that if R is a commutative main ideal ring without zero divisors, then the condition coker A = coker B implies the equivalence of matrices A and B. If R is a noncommutative principal ideal ring without zero divisors, then the implication is not always true [186]. Although, as shown by Nakayama [195], if R is a principal ideal ring without zero divisors, then the elementary divisors of the matrix (i.e. the diagonal elements of the canonical diagonal form of the matrix) are defined uniquely up to similarity. At the same time, as observed in [186], the notion of full divisor gives the canonical form for the cokernel of a matrix, but not for a class of equivalent matrices. As shown by Levy and Robson [186], for semiperfect rings, the following implication holds: if the cokernels of the matrices are isomorphic, then the matrices are equivalent. Huralnik [153] showed that over any ring of stable rank 1 this implication also takes place. The notion of stable rank came to the ring theory from the K-theory and proved to be useful for solving a number of open problems in the ring theory. It was first introduced by Bass [94]. A row (a 1,..., a n ) of elements of R is called right unimodular if there are elements x 1,..., x n R such that a 1 x 1 + a n x n = 1. The stable rank of R (abbreviated as st.r. (R)) is the smallest positive integer n for which following condition holds: for any right unimodular row (a 1,..., a n, a n+1 ) of elements of R there

17 PREFACE 17 exist elements b 1,..., b n R such that the row (a 1 +a n+1 b 1, a 2 + a n+1 b 2,..., a n +a n+1 b n ) is right unimodular. If there is no such naturall n, the stable rank of R is considered to be equal infinity (st.r. (R) = ) [233, 237]. Kaplansky [159] indicated that a regular ring is unit regular if and only if its stable rank equals 1. Menal, Monkazi [191] have shown that the stable rank of an Hermitian ring does not exceed 2. In [86, 88], it is shown that the stable rank of any separatively regular ring equals either 1, 2 or. Note that the problem of the existence of regular rings of finite stable rank 3 is open [180]. In the same publications [86, 88] it is proved that if a separative regular ring is of finite stable rank, then this ring is an Hermitian ring, and hence a ring which can be diagonalized. It should be noted that examples of simple regular rings with stable rank 1 or are known and that the existence of a simple regular ring of finite stable rank greater than 1 is an open problem [117, 180]. In the paper [137], it is shown that any commutative principal ideal ring without zero divisors of stable rank 1 is Euclidean. Stable rank of different classes of rings is studied in [78, 79, 86, 87, 88, 90, 94, 111, 112, 115, 116, 117, 137, 149, 153, 159, 161, 178, 179, 180, 191, 192, 194, 201, 205, 207, 210, 211, 214, 226, 231, 232, 233, 235, 237]. In [232], examples of rings of stable rank 1 are presented, and the simplest properties of these rings are established. In [106], it is shown that any commutative Bezout ring of stable rank 1 is an elementary divisor ring. Hatalevych [9] extended this result to the case of duo-rings. Stable rank of exchange rings is studied in [86, 88, 90, 111, 116].

18 Chapter 1 PRELIMINARIES In this chapter we will present basic definitions and well known facts which are important in this study, as well as some useful notations and conventions. Throughout the text, the word ring means an associative ring R with nonzero identity element (1 0). All modules will be right unitary modules. Some frequently used notations: U(R) group of units of the ring R; J = J(R) Jacobson radical of R; N = N(R) nilradical of R; R n set of all n - component rows with entries from ring R; n R set of all n - component columns with entries from ring R; R n rind of all n - dimensional square matrices over ring R; GL n (R) = U(R n ) general linear group of ring R; SL n (R) special linear group of ring R; GE n (R) group of elementary matrices of ring R; st.r. R stable rang of ring R; R = R \ {0} set of all nonzero elements of ring R; G.C.D. greatest common divisor; (a, b) greatest common divisor of elements a and b; L.C.M. least common multiply; 18

19 Chapter 1. Preliminaries 19 [a, b] least common multiply of elements a and b; R-mod category of left R-modules; mod-r category of right R-modules; spec R space of prime ideals of commutative ring R; mspec R space of maximal ideals of commutative ring R; min R space of minimal prime ideals of commutative ring R; a b element a divides element b; a b element a is a full divisor of element b, i.e. RbR ar Ra. Now we are going to refresh in mind some necessary definitions, facts and well known results. Definition A right (left) principal ideal ring ring in which every right (left) ideal is principal. [13] is a Definition A right (left) Bezout ring [158] is a ring in which every finitely generated right (left) ideal is principal. A Bezout ring is a ring which is both right and left Bezout ring. A right principal Bezout ring is a ring which is right principal ring and left Bezout ring. Examples of such rings can be found in [125, 127]. Definition A ring R without zero divisors is called a right Ore domain if ar br {0} for any nonzero elements a, b R. Analogously, a ring R without zero divisors is called a left Ore domain if Ra Rb {0} for any nonzero elements a, b R. A ring which is a left and right Ore domain is called an Ore domain. Any Bezout ring without zero divisors is an Ore domain [221]. Throughout our study we need to use different types of element factorization in the rings. So we have to define all this types.

20 20 Chapter 1. Preliminaries Definition An element of a ring is called an atom [73], if it is noninvertible, nonzero and cannot be represented as product of two noninvertible elements. An atomic decomposition of a given element is its decomposition as product of units and atoms [125]. Definition Let R be a ring without zero divisors. A nonzero element a R is called left finite [125], if any proper left divisor of a is either invertible or finite product of atoms; a right finite element is an element such that any right divisor of this element is either invertible or finite product of atoms; finally, an element a is called finite, if it is both left finite and right finite. Definition A nonzero element is called left infinite, if it is noninvertible and any its finite left divisor is invertible. A right infinite element is a nonzero noninvertible element without right atomic divisors. An infinite element is an element which is either left or right infinite [125]. Definition We say that two elements a and b in a ring R are associated, if a = ubv, (1.1) for some invertible elements u, v in R. If in equality (1.1) we put u = 1 (v = 1), then these elements are called right (left) associated [73]. Definition We say that two elements a and a from a ring R which are not zero divisors are similar [73] (we write a a ) provided R/aR = R/a R. In [73] it is proved that this definition is left-right symmetric, i.e if R/aR = R/a R,

21 Chapter 1. Preliminaries 21 then and the isomorphism R/Ra = R/Ra, R/Ra = R/Ra implies the isomorphism R/aR = R/a R. Definition Two atomic decompositions of an element a in product of not zero divisors a = c 1 c n and a = b 1 b m are said to be isomorphic, if n = m and there is a substitution i i of indices 1, 2,..., n such that b i c i [73]. A nonzero element which is not a zero divisor is said to be factorial, if it has finite atomic decomposition and any two its atomic decompositions are isomorphic. The number of elements in atomic decomposition of a factorial element is called its length. After given definitions we want to remark that an element in Bezout ring without zero divisors is finite if and only if it has finite length. In fact, Bezout ring without zero divisors is principal ideal ring if and only if any nonzero noninvertible element has finite length. In other words, it has to be factorial [125]. Definition A nonzero element a in a ring R is called a right (left) invariant (duo-element) [73], if Ra ar (ar Ra) and it is invariant or duo-element, if Ra = ar.

22 22 Chapter 1. Preliminaries Definition A ring is said to be a right (left) duo-ring if every right (left) ideal of this ring is a 2-sided ideal. If a ring is both left and right duo-ring, then it is called a duo-ring. Definition A ring R is called von Neumann regular [149, 197], if for every element a R one can find some nonzero element x R such that axa = a. For the von Neumann regular rings, the following wellknown statement can be proved. Theorem /[149], Th. 1.1, page 1/ The following properties are equivalent for any ring R: 1) R is von Neumann regular; 2) every right (left) principal ideal in R is generated by an idempotent; 3) every finitely generated right (left) ideal in R is generated by an idempotent. Definition A ring R is said to be unit-regular [149], if for any element a R there exists an invertible element u U(R) such that aua = a. Proposition /[159]/ Let R be a unit-regular ring. Then for every element a R there exist idempotents e 1 = e 2 1, e 2 = e 2 2 R and invertible elements u 1, u 2 R such that a = e 1 u 1 = u 2 e 2. Let us denote by F P (R) the class of all finitely generated projective R-modules.

23 Chapter 1. Preliminaries 23 Definition A ring R is called separative, if, for any A, B F P (R), A A = A B = B B implies A = B. It is useful to remark that all unit-regular rings, all right or left χ 0 -continuous von Neumann regular rings [88] and all von Neumann regular rings that are generally comparable rings ([149], Th. 8.16, page 87) are separative rings. Definition We call a module to be a chain module, if all its submodules are linearly ordered by inclusion [82]. A ring R is called a chain ring, if R is direct sum of left chain modules, as well as direct sum of right chain modules [82]. Definition A ring R is called directly finite or Dedekind-finite, if ab = 1 implies ba = 1 for any pair of elements a and b in R [82]. Definition A ring R is said to be right (left) semihereditary, if any right (left) finitely generated ideal in R is projective [82]. Definition A commutative ring R is called a valuation ring [171], if for any elements a and b in R we have either a b or b a. Remark that any valuation ring is a local Bezout ring [73].

24 24 Chapter 1. Preliminaries Definition We say that a commutative ring R is locally countable, if the set of all its maximal ideals is greater than countable, and also any noninvertible element from R that does not belong to the Jacobson radical belongs only to a countable set of maximal ideals [132]. Definition A ring R is called simple [82, 88], if the set of all 2-sided ideals of R consists only of the zero ideal and R itself. Definition A ring R is called a purely infinite ring [178], if R is not a division ring and for any nonzero element a R there exist elements x, y R such that xay = 1. Definition If the lattice of all right (left) ideals of some ring R is distributive, then we say that this ring R is right (left) distributive. [226], Definition A ring R is called a right (left) quasiduo-ring, if any right (left) maximal ideal in R is a 2-sided ideal [226]. The following result is proved in [226] (page 69, Th. 3.49). Theorem If R is a Bezout ring, then the following are equivalent: 1) R is right distributive; 2) R is left distributive; 3) R is a right quasi-duo-ring; 4) R is a left quasi-duo-ring. As it is shown in [178], for the right quasi-duo-rings the following result can be proved.

25 Chapter 1. Preliminaries 25 Theorem If R is a right quasi-duo-ring, then for any element a R from RaR = R it follows that a is invertible in R. In the case of rings where every maximal right ideal is principal (for example, if R is a right principal ideal ring), the reverse statement is true. In other words: Proposition /[178], page 13, statement 6.3/ If any right maximal ideal in a ring R is principal and for every element a R from RaR = R it follows that a is invertible, then R is a right quasi-duo-ring. Definition The L condition sounds as follows: if RaR = R, then a is an invertible element in R. Now, we will mention another property of the right quasiduo-rings that will be sometimes useful in current study. Theorem /[226], page 4, Th. 3.2/ For a ring R the following are equivalent: 1) R is a right quasi-duo-ring; 2) for any r 1, r 2 R from Rr 1 + Rr 2 = R it follows that r 1 R + r 2 R = R; 2) for any x, y R it is true that xr + (yx 1)R = R; 4) for any finite subset S in R the equality RSR = R implies SR = R. Definition A commutative distributive ring is called an arithmetical ring [169, 222].

26 26 Chapter 1. Preliminaries During the following several pages we will define some types of ideals which are important in our study. Definition A right ideal P in a ring R is called a right prime ideal [1, 2], if from arb P, where a, b R, it follows that either a P or b P. Definition A 2-sided ideal P prime [1], if from ab P, where a, b R, it follows that is called completely a P or b P. In the case of commutative rings, the completely prime ideals coincide with the prime ideals. Definition A right ideal P is called weakly prime, if (a + P )R(b + P ) P implies either a P or b P [229]. Definition A ring R is called right (left) almost atomic, if every right (left) maximal ideal is a right (left) principal ideal. A ring R is called almost atomic, if it is both left atomic and right atomic ring [125]. Definition Let R be a commutative ring. A prime ideal P of a ring R is said to be a minimal prime ideal [238], if for any prime ideal N in R from P N it follows that P = N.

27 Chapter 1. Preliminaries 27 The set of all minimal prime ideals in R will be denoted by min R. Definition A ring R is said to be reduced [190], if the nilradical of R coincides with the zero ideal, i.e. there are no nonzero nilpotents in R. Let x R. If we let D(x) = {P min R x / P }. then the set of all D(x) is a base of the Zarisky topology on min R. When we say that min R is compact, then we mean that it is compact in this topology [190]. Definition We say that a module M is flat in the category R mod, if the functor M : R mod mod Z is exact [82]. A right module M is flat if and only if for every set of elements x 1,..., x n M and λ 1,..., λ n R such that n x i λ i = 0, i=1 there exist elements y 1,..., y m M and a ij R, i = 1, 2,..., n, j = 1, 2,..., m, such that 1) n a ij λ i = 0, j = 1, 2,..., m; i=1 2) x i = m y i a ij, i = 1, 2,..., n. j=1 Definition An ideal I is called flat [82], if I is flat as an R-module.

28 28 Chapter 1. Preliminaries Definition We will say that a module M is an exact R-module [83], whenever we have the equality Mr = (0) if and only if r = 0. Definition An ideal I is called exact, if I is exact as an R-module [83]. The following result can be proved for the Bezout rings with compact space of minimal prime ideals. Proposition /[163], statement 2.2, page 12/ If R is a Bezout ring, then 1) dim (Cl(R)) = 0; 2) (a) min (R) is compact; (b) Zd (R) = P, P min R where Cl(R) is the classical ring of fractions of a ring R, and Zd (R) is the set of all zero divisors in R. Definition We call matrices A and B equivalent [172], if there are unimodular matrices P and Q over a ring R of appropriate dimensions such that A = P BQ. Definition A matrix A over a ring R admits a diagonal reduction [172], if it is equivalent to some diagonal matrix. Definition A matrix A over a ring R admits a canonical diagonal reduction [172], if it is equivalent to a diagonal

29 Chapter 1. Preliminaries 29 matrix ε ɛ ɛ r where ε i ε i+1 for any i (1,..., r 1). The elements ε 1,..., ε r are called invariant factors of the matrix A. Definition If every matrix over a ring R admits a diagonal reduction, then it is said that R is diagonalizable or it is a Henriksen elementary divisor ring [159]. Definition If every matrix over a ring R admits a canonical diagonal reduction, then it is said that R is an elementary divisor ring or it is a classical elementary divisor ring [172]. Definition If every 1 2 (2 1) matrix with entries from R admits a diagonal reduction, then it is said that R is a right (left) Hermite ring [172]. If a ring is left and right Hermite, then it is called an Hermite ring. Let R be a right Hermite ring. Then for every row (a, b) with entries from R ther exists a unimodular matrix P GL 2 (R) such that (a, b)p = (d, 0) (1.2) for some element d R.

30 30 Chapter 1. Preliminaries Let ( ) x u P = y v ( ) and P 1 a1 b = 1. r s Than from equation (1.2), ax + by = d, a = da 1, b = db 1, a 1 R + b 1 R = R. Thus, every right Hermite ring is a right Bezout ring. Theorem /[145]/ A commutative ring R is an Hermite ring if and only if for every elements a, b R there exist elements a 1, b 1, d R such that a = a 1 d, b = b 1 d i a 1 R + b 1 R = R. Theorem /[172]/ A ring R without zero divisors is a right Hermite ring if and only if it is a right Bezout ring. Theorem /[85]/ If R is a left Hermite ring and right Bezout ring, then R is a right Hermite ring. From the previous theorem we obtain a natural conclusion: if R is a commutative and right (left) Hermite ring, then it is also a left (right) Hermite ring, i.e. Hermite ring. Another important class of rings we want to mention in our study is the class of rings with zero divisors in the Jacobson radical. Here we have one important result about such rings. Theorem /[172]/ A ring R with zero divisors in the Jacobson radical is right Hermite if and only if it is right Bezout ring.

31 Chapter 1. Preliminaries 31 Here we mention some sufficient conditions for a ring being an elementary divisor ring. Theorem /[172]/ If every 1 2, 2 1, 2 2 matrix with entries in R admits a canonical diagonal reduction, then R is an elementary divisor ring. Theorem /[181]/ A commutative ring R is an elementary divisor ring if and only if every 2 2 matrix over R is equivalent to some diagonal matrix. Theorem /[172]/ A commutative ring R is an elementary divisor ring if and only if it is an Hermite ring and for every elements a, b, c R such that (a, b, c) = 1, there are elements p, q R such that (pa + qb, qc) = 1. Theorem /[181]/ A commutative ring R is an elementary divisor ring if and only if every finitely presented R-module can be decomposed into direct sum of cyclic modules. Among all commutative rings that are elementary divisor rings we have special class of adequate rings. Definition A commutative ring R is called an adequate ring [181], if 1) R is a Bezout ring; 2) for every pair of elements a, b R (a 0) there exists a pair of elements a 1, d R such that (a) a = a 1 d; (b) (a 1, b) = 1;

32 32 Chapter 1. Preliminaries (c) for every noninvertible factor d of d we have (d, b) 1. The following theorem on adequate rings is also important. Theorem /[181], Th. 8, page 364/ Any adequate ring is an elementary divisor ring if and only if it is an Hermite ring. Considering , starting from now we will talk only on adequate Hermite rings. Theorem /[145], Th. 11, page 365/ Any commutative von Neumann regular ring is an adequate ring. As it is proved in [181], the second condition in the definition of adequate ring can be checked for every element (and for zero too!) in a commutative von Neumann regular ring. The same is true for the valuation rings [181]. Theorem /[181]/ Any nonzero prime ideal of an adequate ring without zero divisors is contained in a unique maximal ideal. Theorem /[181]/ Any commutative semihereditary Bezout ring is an Hermite ring. The following group of results can be proved for commutative elementary divisor rings. Theorem /[172]/ Let R be an elementary divisor ring. Then every finitely presented R-module can be decomposed as direct sum of cyclic modules.

33 Chapter 1. Preliminaries 33 Definition [223] Let R be a commutative ring without zero divisors, where for every pair of elements there exists the greatest common divisor (i.e. R G.C.D.). For any chosen nonzero element ψ of the ring R we will define the Lψ condition: if a, b R and also a and b are coprime, then there are elements x, y R such that the elements xa + yb and ψ are coprime, as well as x and ψ are coprime. Theorem /[223], Theorem 3/ Let R be a commutative G.C.D. ring without zero divisors, with Lψ property for every element ψ in R. Moreover, suppose that R satisfies the condition: if a and b are coprime with ψ, then ab is also coprime with ψ. Then every matrix A with entries from R is quasi-equivalent to some diagonal matrix. Definition Let R be a commutative ring without zero divisors and let a be any nonzero element of R. The set S a = {b br + ar = R} is said to be related to S-torsion in the sense of Komarnitskii [147]. The following condition for elements was first mentioned in [17] by Dubrovin: for any element a R there exists an element b R such that RaR = br = Rb. Thus this condition now is called Dubrovin s condition. Rings with Dubrovin s condition have been studied also in [27]. We want to mention the following fact that equivalence of matrices A = (a ij ) and B = (b ij ) with entries in a ring R implies Ra ij R = Rb ij R. i,j i,j

34 34 Chapter 1. Preliminaries In other words: the given above 2-sided ideal is an invariant for equivalence class of equivalent matrices [27]. The following result was proved by Dubrovin: Theorem /[17]/ If R is a semilocal and semiprime Bezout ring, then following are equivalent: 1) R is an elementary divisor ring; 2) R is a ring with Dubrovin condition; 3) R is a finite direct sum of matrix rings over elementary divisor rings without zero divisors (moreover R is a Bezout ring). The notion of full right divisibility was introduced by Cohn, when he studied right principal Bezout rings (it was connected with reduction of matrices over this type of rings) Definition It is said that an element a in a ring R without zero divisors is a full right divisor of an element b (in this case we write a r b) [181], if there exists a right invariant element c such that br cr ar. Theorem /[181], Th 3.6, page 255/ Every matrix over a right principal Bezout ring without zero divisors is equivalent to some matrix of the form ε ɛ ɛ r ,

35 Chapter 1. Preliminaries 35 where the element ε i is full right divisor of the element ε i+1 for each i (1, 2,..., r 1). It is necessary to remark that the rings studied by Cohn [181], are alternative to classical elementary divisors rings using some special approach. Komarnitskii have proposed one intermediate form of reduced matrix, but not yet canonical diagonal form, for commutative rings and Kaplansky s diagonal form. This type of rings was called the rings with almost invariant elementary divisors. Definition A matrix A = (a ij ) over a ring R admits almost invariant diagonal reduction, if it is equivalent to some matrix of the form ε ɛ ɛ r , where the elements ε 1,..., ε r 1 are invariant, but ε r is not necessary invariant and ε i is a left divisor of ε i+1 for each i (1, 2,..., r 1). Definition If every matrix over a ring R admits almost invariant diagonal reduction, then we say that R is a ring with almost invariant elementary divisors [71]. Also we want to note that this class of rings contains all simple elementary divisor rings without zero divisors as well as duo-rings that are elementary divisor rings.

36 36 Chapter 1. Preliminaries Every matrix over distributive ring admits a canonical diagonal reduction only in the case when the given ring is a duo-ring that is an elementary divisor ring. Thus, considering the variety of matrix canonical diagonal forms, some researchers (Henriksen, Menal, Moncasi, Goodearl etc. [87, 116, 159, 191]) study the rings such that every matrix over them can be diagonalized. Theorem /[159]/ Any unit-regular ring is a diagonalizable ring. Menal and Moncasi in [191] have proved the following theorem for the von Neumann regular rings: Theorem For a von Neumann regular ring R the following statements are equivalent: 1) R is a left Hermite ring; 2) R is a right Hermite ring; 3) R is a diagonalizable ring. On the other hand, over chain-rings and separative von Neumann regular rings only square matrices can be diagonalized. Theorem /[87]/ Over a separative von Neumann regular ring R every square matrix can be diagonalized. Theorem /[185]/ Over a semichain ring R every square matrix can be diagonalized. Theorem /[172]/ Let R be a Bezout ring and A be an m n matrix over R. Let A 1 denote m (n + m) matrix where in the first n columns there are the appropriate columns of the matrix A, and other columns are zeros. Then there is a unimodular matrix U of appropriate dimensions such that the matrix A 1 U is triangular.

37 Chapter 1. Preliminaries 37 Among all rings in the class of elementary divisor rings there is a subclass of rings such that every matrix over these rings can be reduces to its canonical diagonal form only using elementary row and column operations. The rings of this type are known as the rings with elementary matrix reduction and they were introduced by the author in [243]. Now, let us define the following types of column (row) operations with a matrix A over a ring R: 1) column (row) switching; 2) column (row) right (left) multiplication by an invertible element; 3) column (row) addition of one column (row) right (left) multiplied by some element to another column (row). As it is well-known [203], the operations mentioned above are in one-to-one correspondence with right (left) multiplication of a matrix A by some elementary matrix. Definition An elementary n n matrix with entries from R is a square n n matrix of one of the types below: 1) diagonal matrix with invertible diagonal entries; 2) identity matrix with one additional non diagonal nonzero entry; 3) permutation matrix, i.e. result of switching some columns or rows in the identity matrix. Definition It is said that a matrix A over a ring R admits elementary diagonal reduction if it can be reduced to the canonical diagonal form using only elementary column and row

38 38 Chapter 1. Preliminaries operations ɛ ɛ ɛ r , where ɛ i is a full divisor of ɛ i+1 for any i (1, 2,..., r 1). Clearly, if an n m matrix A admits elementary diagonal reduction, then for the matrix A one can find elementary matrices P 1, P 2,..., P k GE n (R), Q 1, Q 2,..., Q s GE m (R) such that ɛ ɛ P 1 P 2 P k AQ 1 Q s = ɛ r Definition If over a ring R every matrix admits elementary diagonal reduction, then R is called a ring with elementary matrix reduction. It is clear that ring with elementary matrix reduction is an elementary division ring. But at the same time there are examples of elementary vision rings that are not rings with elementary matrix reduction. For example, such is the principal ideal ring R[x, y]/(x 2 + y 2 + 1) [101].

39 Chapter 1. Preliminaries 39 Among all rings with elementary matrix reduction there are special subclasses of elementary principal rings. Definition A ring R is called right elementary principal, if for every pair of elements a, b from R there exist an element d R and a matrix P GE 2 (R) such that (a, b)q = (d, 0). Analogously, we say that a ring R is left elementary principal, if for every pair of elements a, b R there exist an element c R and a matrix Q GE 2 (R) such that Q ( ) a b ( ) c =. 0 Obviously, an elementary principal ring is a ring that is both left and right elementary principal ring. Clearly, every ring with elementary matrix reduction is an elementary principal ring. Also, the class of all commutative elementary principal rings coincides with that of quasi-euclidean rings studied by Bougaut [101]. Definition We say that a given commutative ring R has quasi-algorithm, if a function ϕ: R R W (where W is some ordinal) is given such that for any a, b R (b 0) one can find elements q, r R such that a = bq + r and ϕ(b, r) < ϕ(a, b). If one can find some quasi-algorithm on R then this ring is called quasi-euclidean. Natural examples of quasi-euclidean rings are the Euclidean rings, valuation rings and von Neumann regular rings [101].