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Home , Lattice (order), Ordered ring

Lattice-ordered Rings and Modules -ordered Rings and Modules

Stuart A. Steinberg Toledo, OH, USA Stuart A. Steinberg Department of University of Toledo Toledo, OH 43601 USA [email protected]

ISBN 978-1-4419-1720-1 e-ISBN 978-1-4419-1721-8 DOI 10.1007/978-1-4419-1721-8 Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2009940319

Mathematics Subject Classification (2010): 06F25, 13J25, 16W60, 16W80, 06F15, 12J15, 13J05, 13J30, 12D15

© Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com) To Diane Stephen, David, and Julia Preface

A lattice-ordered is a ring that is also a lattice in which each additive translation is order preserving and the product of two positive elements is positive. Many ring constructions produce a ring that can be lattice-ordered in more than one way. This text is an account of the algebraic aspects of the theories of lattice-ordered rings and of those lattice-ordered modules which can be embedded in a product of totally ordered modules—the f -modules. It is written at a level which is suitable for a second-year graduate student in mathematics, and it can serve either as a text for a course in lattice-ordered rings or as a monograph for a researcher who wishes to learn about the subject; there are over 800 exercises of various degrees of difficulty which appear at the ends of the sections. Included in the text is all of the relevant background information that is needed in order to to make the theories that are developed and the results that are presented comprehensible to readers with various backgrounds. In order to make this book as self-contained as possible it was necessary to in- clude a large amount of background material. Thus, in the first chapter we have con- structed the Dedekind and MacNeille completions of a (poset) and developed enough of so that we can present Birkhoff’s char- acterization of a variety and so that we can also verify the existence of free objects in a variety of algebras. Much of the material on lattice-ordered groups (`-groups) in the second chapter appears in those books devoted to the subject. What is new in this book is the emphasis on `-groups with operators. This allows for the common development of basic results about `-groups, f -rings, and f -modules. The Amitsur- Kurosh theory of radicals is developed for the class of `-rings in the second section of Chapter 2 . Still more background material is given in the first two sections of Chapter 4 where the injective hull of a , the Utumi maximal right quotient ring, and the ring of quotients and the module of quotients with respect to a hered- itary torsion theory are constructed and studied for a ring which is not necessarily unital. Also, the Artin–Schrier theory of totally ordered fields is given in the first section of Chapter 5, and enough of the theory of valuations on a field is presented in the second section so that a complete proof of the Hahn embedding theorem for a well-conditioned commutative lattice-ordered can be given.

vii viii Preface

Chapters 3, 4, 5, and 6 constitute the heart of the book. While not every known result on the topics included is presented, enough is presented so as to make the text, by which I mean the exercises also, reasonably complete. The first section of Chap- ter 3 develops the basic theory of `-rings including the fact that canonically ordered matrix rings have no unital f -modules. Section 4 shows that the fundamental process of embedding an f -algebra in a unital f -algebra is more complicated than the anal- ogous embedding for algebras and cannot always be carried out. The fifth section shows how to construct power series type examples of `-rings and `-modules using a poset which is a partial and which is rooted in the sense that the set of upper bounds of each element is a chain. The basic structure of f -rings is given in the third section of Chapter 3 and some of the richer structure of archimedean f -rings is given in the sixth section. In the last two sections the structure of `-rings in other varieties is examined. The seventh section studies those `-rings that have squares positive and gives conditions on a partially ordered generalized semigroup for the lexicographically ordered semigroup ring to have this property. The last sec- tion considers those `-rings which satisfy polynomial constraints more general than that of squares being positive. One effect of these constraints is to coalesce the set of nilpotent elements into a subring or an and to force an `-semiprime ring to lack nilpotent elements. Also included in this section is a proof of the commutativity of an archimedean almost f -ring. Chapter 4 concentrates on the category of f -modules. The most conclusive re- sults occur for a semiprime f -ring whose maximal right quotient ring is an f -ring extension and whose Boolean algebra of annihilators is atomic. In the third section necessary and sufficient conditions for the module (or ring) of quotients to be an f -module (or an f -ring) extension are given, and the structure of right self-injective f -rings is given. The unique totally ordered right self-injective ring that does not have an is exhibited. The module and order theoretic properties that determine when a nonsingular f -module is relatively injective are given in the fourth section—there are no injectives in this category of f -modules. A useful rep- resentation of the free nonsingular f -module is given in the last section and the size of a disjoint set in a free f -module is determined. In a totally ordered field the set of values—those convex subgroups which are maximal with respect to not containing a given nonzero element—becomes a to- tally ordered under the induced on it by multiplication in the field. A proof of the Hahn Embedding Theorem for totally ordered fields is given in the second section of Chapter 5; namely, a totally ordered field is embedded in a power series field where the exponents belong to this value group of the field and the co- efficients are real numbers. Also, a totally ordered is embedded in a totally ordered division algebra over the reals. In the third section of Chapter 5 the Hahn Embedding Theorem is given for a lattice-ordered commutative domain which satisfies a finiteness condition, and another embedding theorem for a suitably con- ditioned `-field into a formal power series crossed product `-ring is given. Also, the theory of archimedean `-fields is presented and lattice orders other than the usual are constructed for the field of real numbers. Preface ix

Chapter 6 begins with a characterization of the canonically ordered real semi- group `-algebra over a locally finite left cancellative semigroup, and semigroup `- rings in which squares are positive are studied in more detail. In the second section it is shown that in an `-algebra in which the nonzero f -elements are not zero- each algebraic f -element is central. A complete description is also given of those rings which have the property that each partial order is contained in a total order. In the third section more commutativity theorems are presented. It is shown that a totally ordered domain which is co-`-simple and which has a positive semidefinite form with a nontrivial solution must be commutative. A similar conclusion giving the centrality of f -elements appears for an `-ring in which the commutators are suit- ably bounded. A proof of Artin’s solution to Hilbert’s 17th problem which is mainly dependent on the variety of f -rings generated by the real numbers is also included in this section. Lattice orders on the n×n matrix algebra over a totally ordered field are considered in the last section. When the field is archimedean or when n = 2 all of the lattice orders are described, and it is shown that in these cases the usual order is the only lattice order in which the identity element is positive. A few words about the method of referencing are in order. An exercise is ref- erenced by its number alone when the exercise occurs in the section in which it is referenced, and, otherwise, it is referenced by its number preceded by the chapter and section numbers in which it occurs. A reference to a theorem or a numbered line uses all three of its numbers. Paul Taylor’s package was used in preparing the diagrams in the text. I wish to thank Joanne Guttman and Shirley Michel for their splendid job of typing and prepa- ration of the manuscript. I would also like to thank my colleague Charles Odenthal for making the diagrams fit.

Toledo, Ohio Stuart Steinberg August 17, 2009 Contents

Preface ...... vii

List of Symbols ...... xiii

1 Partially Ordered Sets and Lattices ...... 1 1.1 Partially Ordered Sets ...... 1 1.2 Lattices ...... 10 1.3 Completion ...... 16 1.4 Universal Algebra ...... 21

2 Lattice-ordered Groups ...... 33 2.1 Basic Identities and Examples ...... 34 2.2 and ...... 43 2.3 Archimedean `-groups ...... 54 2.4 Prime Subgroups, Representability, and Operator Sets ...... 88 2.5 Values ...... 96 2.6 Hahn Products and the Embedding Theorem ...... 113

3 Lattice-ordered Rings ...... 125 3.1 Basics, Examples, and Nonexamples ...... 126 3.2 Radical Theory ...... 143 3.3 f -Rings ...... 167 3.4 Embedding in a Unital f -Algebra ...... 179 3.5 Generalized Power Series Rings ...... 200 3.6 Archimedean f -Rings ...... 219 3.7 Squares Positive...... 233 3.8 Polynomial Constraints ...... 251

4 The Category of f -Modules ...... 281 4.1 Rings of Quotients and Essential Extensions ...... 281 4.2 Torsion Theories and Rings of Quotients ...... 307 4.3 Lattice-ordered Rings and Modules of Quotients ...... 332

xi xii Contents

4.4 Injective f -Modules ...... 364 4.5 Free f -Modules ...... 391

5 Lattice-ordered Fields ...... 419 5.1 Totally Ordered Extensions of Ordered Fields ...... 419 5.2 Valuations and the Hahn Embedding Theorem ...... 431 5.3 Lattice-ordered Fields ...... 475

6 Additional Topics ...... 511 6.1 Lattice-ordered Semigroup Rings ...... 511 6.2 Algebraic f -Elements Are Central ...... 535 6.3 More Polynomial Constraints on Totally Ordered Domains ...... 553 6.4 Lattice-ordered Matrix Algebras ...... 575

Open Problems ...... 609

References ...... 613

Index ...... 623 List of Symbols

|X|, card (X) cardinality of X ⊆, ⊇ inclusion ⊂, ⊃ proper inclusion ∪˙ disjoint P(X) of X; P-radical of the `-ring X ΠXi of {Xi :∈ I} (xi)i∈I an element of ΠXi XI (Xn) cartesian product of I (n) copies of X X∗ set of nonzero elements in the or ring X; annihilator of X in a reduced ring Pog (Poag) category of (abelian) po-groups Log (Loag) category of (abelian) `-groups MR (u-MR) category of (unital) right R-modules po-( f -, `-)MR category of right po-modules ( f -modules, `-modules) over R ns f -MR category of nonsingular right f -modules over R U (R) group of units in the monoid R GL(n,K) group of units in the n × n matrix ring over K ord(T) ordinal number of the well-ordered set T W(α) initial segment of the ordinals determined by α ℵα infinite cardinal number ωα first ordinal whose cardinality is ℵα U(X),UP(X) set of upper bounds of X (in P) L(X),LP(X) set of lower bounds of X (in P) Ls(X) (Us(X)) set of strict lower (upper) bounds of X lubX, lubPX least upper bound of X (in P) supX, sup X least upper bound of X (in P) W P i∈I xi least upper bound of {xi : i ∈ I} a ∨ b least upper bound of {a,b} glbX, glbPX greatest lower bound of X (in P) infX, inf X greatest lower bound of X (in P) V P i∈I xi greatest lower bound of {xi : i ∈ I}

xiii xiv List of Symbols a ∧ b greatest lower bound of {a,b} a||b a is incomparable to b [a,b] closed interval; commutator X ≤ Y x ≤ y for each x ∈ X and each y ∈ Y P←× Q antilexicograhically ordered product P→× Q lexicographically ordered product M(P) MacNeille completion of the poset P D(P) Dedekind completion of the poset P; subring generated by the d-elements on an `-module P or in an `-ring P lim direct limit −→ lim inverse limit ←− C [A,B], [A,B] from A to B in the category C Ω operator domain Ω(n) set of n-ary operators ω(a1,...,an) evaluation of the n-ary operation ω at a1,...,an ωA a constant in the Ω-algebra A SΩ (X) subalgebra of an Ω-algebra generated by X A (Ω) the class (category) of Ω-algebras W(Ω,X) the Ω-row algebra on X FΩ (X),F(X) Ω-word algebra on X, free Ω-algebra on X ∗ VΩ (S,X),S variety of Ω-algebras satisfying the identities in S on the alphabet X V (C ) variety of Ω-algebras generated by the class C VC(R) (V (R)) variety of C-`-algebras (`-rings) generated by R C ∗ set of identities satisfied by each Ω-algebra in the class C G+ set of positive elements in the po-group G ¢i∈IGi direct sum of the groups, rings, or modules Gi ⊕i∈IGi direct sum of the po-groups, po-rings, or po-modules Gi Aut(A) automorphism group of A End(G)+ set of po-homomorphisms of the po-group G x+ (x−) positive (negative) part of x |x| of x d(M) divisible hull of the M; set of d-elements on the module M d(P,S) divisible closure of the partial order P in S dr(R) set of right d-elements in the ring R V(Γ ,Gγ ) Hahn product of the po-groups Gγ indexed by the poset Γ Σ(Γ ,Gγ ) subgroup of elements in the Hahn product whose support is finite W(Γ ,Gγ ) subgroup of elements in the Hahn product whose support is a W-set Dα domain of the function α; right inner derivation determined by the ring element α supp α support of the function α List of Symbols xv

S(α) closure of the support of α A− (A◦) closure (interior) of A in a topological R− R ∪ {∞,−∞} Nε (x) ε-neighborhood of x lim xn limit of a convergent net N (x) neighborhood system of x C(X) (C∗(X)) set of continuous (bounded) real-valued functions on the topological space X E(X) set of continuous extended real-valued functions on X D(X) set of elements in E(X) which are real-valued on a dense of X C(X) (CG(X)) convex `-subgroup (of G) generated by X G CΩ (X) (CΩ (X)) convex `-Ω-subgroup (of G) generated by X [X] `-subgroup generated by X R R(1) CC(1) for the `-subalgebra R of a unital C-`-algebra M(1) set of elements in R(1) which are infinitely smaller than 1 with respect to C C (G) (CΩ (G)) lattice of convex `-subgroups (convex `-Ω-subgroups) of the (Ω-)`-group G A⊥ (A⊥R ) polar of A (in R) B(G) Boolean algebra of polars of the `-group G Ge c`-essential closure of the archimedean `-group G lex B lexicographic extension of the convex `-subgroup B AΩ largest convex `-Ω-subgroup contained in A trunk (P) trunk of the poset or `-group P ΓΩ (a) (ΓΩ (a,G)) set of Ω-values of a (in G) Γ (a) (Γ (a,G)) set of values of a (in G) Γ (G) (ΓΩ (G))(Ω-)value set of the `-group (Ω- f -group) G Spec(R) set of prime ideals of the ring or R Rn n × n matrix ring over the ring R R[x1,...,xn] polynomial ring over R in the noncommuting or commuting indeterminates x1,...,xn D[x;σ,δ] left skew polynomial ring D[δ,σ;x] right skew polynomial ring R[X]0 polynomials over R in the set of indeterminates X with zero constant term P(a1,...,an) set of polynomials which are positive when evaluated at (a1,...,an) ∗ P (a1,...,an) set of nonconstant polynomials in P(a1,...,an) p0(x) derivative of a polynomial f (M) (F(M) set (subring) of f -elements on the `-module or `-ring M fr(R) (Fr(R)) set (subring) of right f -elements in the `-ring R f¯(M) (F¯(M)) set (subring) of elements multiplication by which are f -maps on the `-module or `-ring M fr(∆) ( f`(∆)) set of elements in the pops ∆ right (left) translation by which xvi List of Symbols

preserves incomparability with respect to any given element hXi (hXir) (right) `-ideal generated by X XA subgroup generated by all xa with x in X and a in A A[n] `-ideal generated by An r(X;R) (r(X)) right annihilator of X in R `(A;M) (`(A)) left annihilator of A in M Ann(R) Boolean algebra of annihilator ideals of a semiprime ring r`(X;R) (r`(X)) right `-annihilator of X in R ``(A;M) (``(A)) left `-annihilator of A in M i(V) elements in an F-vector lattice V which are F-infinitely smaller than each F-strong order unit SP class of P-semisimple `-rings UN upper radical determined by the class of `-rings N LA lower radical determined by the class of `-rings A `-β lower `-nil radical `-Nil upper `-nil radical `-Ng generalized `-nil radical R(R) of the modular maximal right `-ideals of the `-ring R O(R) intersection of the right `-primitive `-ideals of the `-ring R + J (Jle ft , J ) (left, positive) Johnson radical of the `-ring R N(R) (Nn(R)) set of nilpotent elements (of index at most n) in the ring R M(R) set of elements in the `-ring R whose absolute values are nilpotent S (V ) class of `-rings which, modulo their lower `-nil radical, belong to the variety V S ( f ) S (V ) for V the variety of f -rings SP (V ) class of `-rings which, modulo their P-radical, belong to the variety V Ru unital cover of the f -ring R RC-u C-unital cover of the f -algebra R x ¿A y (x ¿ y) x is infinitely smaller than y with respect to A (Z) ∑(A ∗ ∆) crossed product V(A ∗ ∆) formal power series crossed product Nn(∆) set of elements α in the pops ∆ for which nα is not defined E (M) set of essential submodules of M t(M) torsion submodule of M (Y : X) set of ring elements r with Xr ⊆ Y Ds (R : s) E(M) injective hull of the module M D(R) set of dense right ideals of the ring R Z(M);Z(R) singular submodule of the module M, center of the ring R Zr(R) right singular ideal of the ring R c`(N) closure of the submodule N Q(R) (Qr(R)) maximal right quotient ring of the ring R List of Symbols xvii

Q2(R) maximal two-sided quotient ring of R Q(P) localization of the positive cone P of a field obtained by inverting the elements of P ≤q partial order determined by Q(P) ≤u usual total order of R (T ,F ) a torsion theory tT (M) sum of the T -submodules of M Tt (Ft ) torsion class (torsion-free class) determined by the left exact radical t F (Ft ) right (determined by the left exact radical t) tF radical determined by the topology F G Goldie topology MF, Q(M),(MΣ ) module of quotients of the module M (with respect to the multiplicatively closed set Σ) Qc(R) (RΣ ) classical right quotient ring of R (with respect to the multiplicatively closed set Σ) c`F(A) F-closure of the submodule A CF(M) lattice of those submodules of M for which the quotient is F-torsion-free C (M) (Cr(R)) CG(M) (C (RR)) M ⊗R N tensor product of the modules M and N po M ⊗R N po-tensor product of the po-modules M and N ` M ⊗R N `-tensor product of the po-modules M and N `` M ⊗R N `-tensor product of the `-modules M and N FM free representable f -module over the po-module M in a category C Fn free nonsingular f -module of rank n ΓD (UD) value group (group of units) of the valued division ring D RD (JD) (maximal ideal of the) valuation ring of the valued division ring D Fˆ Cauchy completion of the totally ordered division ring F Ga the largest totally ordered subgroup of the archimedean `-group G which contains a > 0 Cn(S,K+∗) group of positive n-cochains of the semigroup S with coefficients in the totally ordered field K Z2(S,K+∗) group of positive 2-cocycles of S over K B2(S,K+∗) group of positive coboundaries of S over K H2(S,K+∗) second cohomology group of S over K f ∗ on cohomology groups induced by the semigroup homomorphism f

∆0, ∆Γ1,Γ2 new pops obtained by modifying the partial orders of the pops ∆ P0 (PΓ1,Γ2 ) Hahn ordering of the generalized semigroup ring A[∆0] (A[∆Γ1Γ2 ]) Pn,α,β (Pn,α ) partial orders of a generalized semigroup ring tr a (det a) trace (determinant) of the matrix a at transpose of the matrix a ( j) a (a( j)) jth column (row) of the matrix a xviii List of Symbols d(δ1,...,δn) diagonal matrix with diagonal entries δ1,...,δn + t P(a) the partial order (K )na of the matrix ring Kn over the po-ring K Irr(a,K) irreducible polynomial of the algebraic element a over the field K