Table Of ContentNon-Commutative Valuation Rings and
Semi-Hereditary Orders
K-Monographs in Mathematics
VOLUME 3
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The titles published in this series are listed at the end of this volume.
Non-Commutative
Valuation Rings and
Semi-Hereditary Orders
by
Hidetoshi Marubayashi
Department ofM athematics.
Naruto University ofE ducation.
Naruto. Japan
Haruo Miyamoto
Department ofM athematics.
Anan College of Technology.
Anan.Japan
and
Akira Ueda
Department ofM athematics.
Shimane University.
Matsue. Japan
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4853-0 ISBN 978-94-017-2436-4 (eBook)
DOI 10.1007/978-94-017-2436-4
Printed on acid-free paper
All Rights Reserved
© 1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
PREFACE
CHAPTER I. SEMI-HEREDITARY AND PRUFER ORDERS
§1 . Elementary properties of orders 1
§2. General theory of semi-hereditary and Prufer orders 7
§3. The centers of semi~hereditary D-orders 12
§4. Characterizations of semi-hereditary D-orders 15
CHAPTER II. DUBROVIN VALUATION RINGS
§5. Elementary properties of Dubrovin valuation rings 21
§6. The ideal theory of Dubrovin valuation rings 28
§7. Dubrovin valuation rings of a simple Artinian ring with
finite dimension over its center 35
§8. Invariant and total valuation rings of a division ring with
finite dimension over its center 41
§9. The existence and conjugacy theorems 49
§1O. The residue rings and value groups 53
§1 1. Immediate compatible extensions 57
§12. Dubrovin valuation rings integral over their centers 70
§1 3. Prime and primary ideals of Dubrovin valuation rings 74
CHAPTER III; SEMI-LOCAL BEZOUT ORDERS
§14. Localizations of Bezout orders 79
§15. Approximation theorem for Dubrovin valuation rings 83
§16. The intersection properties of semi-local Bezout orders 89
§17. Characterizations of semi-local Bezout orders 98
§18. Prime and primary ideals of semi-local Bezout orders 103
§19. Defect theorem for central simple algebras 107
CHAPTER IV. THE APPLICATIONS AND EXAMPLES
§20. Idealizers of semi-hereditary V-orders 109
§2I. Prufer orders finitely generated over their centers 120
§22. Strongly Prufer orders 124
§23. Value functions on simple Artinian rings 133
§24. Dubrovin valuation rings in crossed product algebras 141
§25. Matrix rings over invariant valuation rings 154
§26. Bezout orders and Henselization 161
APPENDIX
AI. Semi-perfect rings and serial rings 174
A2. Coherent rings 175
A3. Azumaya algebras 177
A4. The lifting idempotents 178
AS. Wedderburn's Theorem 181
VI
REFERENCES
183
INDEX OF NOTATION 188
INDEX 190
PREFACE
Much progress has been made during the last decade on the subjects of non
commutative valuation rings, and of semi-hereditary and Priifer orders in a simple
Artinian ring which are considered, in a sense, as global theories of non-commu
tative valuation rings. So it is worth to present a survey of the subjects in a
self-contained way, which is the purpose of this book.
Historically non-commutative valuation rings of division rings were first treat
ed systematically in Schilling's Book [Sc], which are nowadays called invariant
valuation rings, though invariant valuation rings can be traced back to Hasse's
work in [Has]. Since then, various attempts have been made to study the ideal
theory of orders in finite dimensional algebras over fields and to describe the Brauer
groups of fields by usage of "valuations", "places", "preplaces", "value functions"
and "pseudoplaces".
In 1984, N. 1. Dubrovin defined non-commutative valuation rings of simple
Artinian rings with notion of places in the category of simple Artinian rings and
obtained significant results on non-commutative valuation rings (named Dubrovin
valuation rings after him) which signify that these rings may be the correct def
inition of valuation rings of simple Artinian rings. Dubrovin valuation rings of
central simple algebras over fields are, however, not necessarily to be integral over
their centers. If we extend the class of rings to semi-local Bezout orders, then we
see that there always exist semi-local Bezout orders in central simple algebras over
fields which are integral over the centers. This fact shows that semi-local Bezout
orders are very important class of rings from the viewpoint of orders. In fact, we
have fruitful results on semi-local Bezout orders as it is seen in Chapter III.
In commutative ring theory, a domain D is Priifer if and only ifthe localization
of D at any maximal ideal is a valuation ring. This means that Priifer domains
are considered as a global theory of valuation rings. Furthermore, an ideal of D is
projective if and only if it is invertible, equivalently a progenerator of the category
of D-modules.
In non-commutative ring theory, "progenerators" implies, of course, "projec
tivities". However the converse is not necessarily true. This suggests us that there
are, at least, two classes of orders which may be considered as global theories of
valuation rings, that is, semi-hereditary orders and Priifer orders which are based
on projectivities and progenerators, respectively.
This book consists of four chapters. Chapter I contains the preliminary ma
terial on orders, the general theory of semi-hereditary and Priifer orders. The
several characterizations of semi-hereditary orders integral over the centers are
also given here, which are applicable to Dubrovin valuation rings and semi-local
Bezout orders later on.
The elementary properties of Dubrovin valuation rings are summarized in
Chapter II including some results on invariant valuation rings and total valuation
rings, some of which have their own particular properties and are needed for the
study of Dubrovin valuation rings.
Chapter III is devoted to the study of semi-local Bezout orders in which main
tools are localizations and the intersection properties. The approximation theorem
viii
for Dubrovin valuation rings is also described in this chapter.
Much of the results and the ideas in Chapters I - III are applied to the further
development of Dubrovin valuation rings, Priifer orders and to give some examples
of semi-hereditary (maximal) orders. These are summarized in Chapter IV which
contains the following; the tensor products of Oubrovin valuation rings, Oubrovin
valuation rings in crossed product algebras, the idealizers of semi-hereditary orders,
matrix rings over invariant valuation rings which are semi-hereditary maximal
orders and Henzelizations of Bezout orders.
For convenience of the reader, we make a brief appendix comprising some old
and famous theorems not appeared in any book in the references and also some
results on weak global dimensions used in §4 with complete proofs.
The reader is expected to be familiar with basic facts from non-commutative
ring theory, commutative valuation theory and homological algebras which are
quoted from the excellent books in the references.
We add some remarks which indicate the sources of the contents and open
problems in the end of each section for facilitating the further research of the
reader.
The main materials in this book are taken from the following; Brungs and
Grater [BGI], [BG2] ; Oubrovin [D2], [03], [D4]; Grater [G5], [G7]; Haile and
Morandi [HM]; Haile, Morandi and Wadsworth [HMW] ~ Morandi [M2], [M3],
[M4] ; Wadsworth [WI], [W3]; the authors [MMI], [MM2], [MM3] , [MUI]. In
several places, we give shorter proofs than ones in the original papers and different
approaches from the original papers, which make the book as self-contained and
concise as possible. However, without their excellent works mentioned above,
this book might never have been written. It is our pleasure to thank them for
their great contributions to the subjects. We also thank Prof. A. Bak for giving
us valuable suggestions on the book, Prof. J. Alajbegovic for recommending us
to publish the book from Kluwer Academic Publishers and the staff of Kluwer
Academic Publishers for their care and helpfulness.
CHAPTER I
SEMI-HEREDITARY AND PRUFER ORDERS
§1. Elementary Properties of Orders
Throughout this book, a ring is always an associative ring with identity element
1 (except for Appendix). In this section, we shall give a few definitions, notations
and elementary properties of orders. For a ring R, we denote by U(R) the set
of all units of R and by CR(O) the set of all regular elements (that is, non-zero
divisors) of R (or C(O) if there are no confusions).
Let C be a multiplicatively closed subset of R. Then we say that R satisfies
the right Ore condition with respect to C or that C is a right Ore set of R if, for any
a E Rand c E C, there exist bE Rand dEC such that ad = cb. If C ~ C(O), then
it is called a regular right Ore set of R. A {regular} left Ore set of R is defined
similarly. If C is a (regular) right and left Ore set of R, then it is simply called a
{regular} Ore set of R.
If C is a regular right Ore set of R, then we can construct the quotient ring
of R, that is, an overring T of a ring R is called the right quotient ring of R with
respect to a regular right Ore set C if
(i) any element c of C is a unit of T and
(ii) for any q E T, there exist a E Rand c E C such that q = ac-1•
We denote the ring T by Re. We note that for a multiplicative subset C of R with
C ~ CR(O), the right quotient ring Rc of R with respect to C exists if and only if
R satisfies the right Ore condition with respect to C (cf. [MeR, Chap. 2]).
Now let R be a subring of a ring Q. If Q is the right quotient ring of R with
respect to CR(O), then we call R a right order in Q and sometimes denote the ring
Q by Q(R). In particular, R is a right order in Q if and only if R satisfies the
right Ore condition with respect to CR (O). A left order in Q is defined similarly
and a ring which is both a right and a left order in Q is called an order in Q.
Let R be a ring and let M be a right R-module. An R-submodule L of M
is said to be essential if LnN # 0 for any non-zero R-submodule N of M. Using
Zorn's lemma, we can show that, for any R-submodule L of a right R-module M,
=
there exists an R-submodule L' of M such that LnL' 0 and LffiL' is essential in
M (cf. [MeR, Lemma 2.2.2 (v)]). If a right ideal I of R is an essential R-submodule
of R, then I is called an essential right ideal. A right R-module U is said to be
uniform if, for any non-zero R-submodules U1 and U2 of U, U1 n U2 # 0, that
is, any non-zero R-submodule of U is an essential R-submodule of U. A right
R-module M is said to have finite Goldie dimension if it contains no infinite direct
sum of non-zero R-submodules.
For any subset X of a ring R, we set rR(X) = {a E R I Xa = O} and call it
the right annihilator of X. If there are no confusions, then we denote it by r(X).
The left annihilator £(X) of X is defined similarly.
A ring R is called a right Goldie ring if R satisfies the ascending chain condi
tion on right annihilators and has finite Goldie dimension as a right R-module. A
H. Marubayashi et al., Non-Commutative Valuation Rings and Semi-Hereditary Orders
© Springer Science+Business Media Dordrecht 1997
2 Chapter I
left Goldie ring is defined similarly and we call R a Goldie ring if R is a right and
left Goldie ring. Here we summarize elementary properties of Goldie rings which
are frequently used in this book as follows.
Theorem 1.1. Let R be a ring.
(1) The following are equivalent:
(i) R is a (semi-)prime right Goldie ring.
(ii) R has a right quotient ring Q which is (semi-)simple Artinian, that is, R
is a right order in a (semi-)simple Artinian ring Q.
(2) Suppose that R is a semi-prime right Goldie ring and let I be a right ideal of
R.
(i) I is essential if and only if I contains a regular element of R.
(ii) If I is essential, then I is generated by regular elements of R.
(3) If a right R-module M has finite Goldie dimension, then
(i) M contains a finite direct sum of uniform R-submodules which is an
essential R-submodule, and
(ii) if U1 Ef) .•• Ef) Un is a direct sum of uniform R-submodules of M which is
essential in M, then n is independent of the choice of the Ui. We call n
the Goldie dimension of M and denote it by dR(M) or d(M) if there are
no confusions.
(4) If R is a semi-prime right Goldie ring with right quotient ring Q and U is
a uniform right ideal of R, then EndR(U) is a right order in the division
ring EndQ(UQ), where EndR(U) is the endomorphism ring of U as a right
R-module.
Proof. See [McR, Lemma 2.2.8, Proposition 2.3.5, Theorems 2.2.9, 2.3.6 and 3.3.5
and Corollary 3.3.7J. 0
For any ring R, we denote by Z(R) the center of R. In the case of algebras,
we sometimes consider orders in an Artinian ring with extra conditions: Let Q be
an Artinian ring and let F be a finite direct sum of fields such that Q is a finitely
generated F-module and Z(Q) ;2 F. A subring R of Q is called an order in Q
if FR = Q and F = Q(D), the quotient ring of D, where D = RnF (of course,
Q is the quotient ring of R). Furthermore, if R is integral over D, that is, any
element of R is integral over D, then R is called a D-order in Q. AD-order S in
Q is called maximal if T is a D-order in Q with T 2 S, then T = S. Let R be a
D-order. Then, note that there exists a maximal D-order in Q containing R by
Zorn's lemma. In this book, we shall study aD-order R in a simple Artinian ring
Q with finite dimension over F = Z(Q). We only need the general case, that is,
F is a finite direct sum of fields and Q is an Artinian ring when we consider the
factor rings of orders.
Let R be an order in a ring Q. Then a right R-submodule I of Q is called a
right R-ideal of Q if
(i) InU(Q):f. 4> and
(ii) there exists c E U(Q) such that cl ~ R.
