JOURNAL OF ALGEBRA 198, 412᎐427Ž. 1997 ARTICLE JA977145

The Real Spectrum of a Noncommutative Ring

Ka Hin LeungU

National Uni¨ersity of Singapore, 119260, Singapore

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Murray Marshall† and Yufei Zhang‡

Uni¨ersity of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada

Communicated by Susan Montgomery

Received September 30, 1996

Spaces of orderings were introduced by the second author in a series of paperswx 15᎐19 and various structure results were obtained generalizing results proved earlier for formally real fields by various people: E. Becker, L. Brocker,¨ R. Brown, T. Craven, A. Prestel, and N. Schwartz. Inwx 6, 26 Žalso seewx 20, 23, 25. Craven and Tschimmel prove that formally real skew fields give rise to spaces of orderings. Inwx 8Ž also seewx 9. Kalhoff extends this to ternary fieldsŽ. also called planar ternary rings . Inwx 10, 11Ž also see wx28. Kleinstein and Rosenberg and Knebusch, respectively, show how spaces of orderings arise in the study of realŽ. commutative semi-local rings. Orderings on commutative rings have been studied extensively since the introduction of the real spectrum by Coste and Roy in the early 1980s.

* E-mail address: [email protected]. † The research of the second author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. E-mail address: [email protected] ask.ca. ‡ E-mail address: [email protected].

412

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 413

Here, the main motivation comes from real algebraic geometry and the study of semialgebraic sets; seewx 2, 12 . Abstract real spectraŽ also called spaces of signs. were introduced just recently inwx 1, 4, 21 . Orderings on a real commutative ring give rise to an abstract real spectrum. The axioms of an abstract real spectrum generalize those of a space of orderings. Con- versely, if Ž.X, G is an abstract real spectrum then G has prime ideals and Ž. at each prime ᒍ : G we can form the so-called residue space Xᒍᒍ, G which is a space of orderings. Inwx 1, 21 various local-global principles are proved for abstract real spectra. In particular, the results on minimal generation of semialgebraic sets due to L. Brocker¨ and C. Scheiderer carry over to this abstract setting. Ordered skew fields were considered already by D. Hilbert in connec- tion with his work on the foundations of geometry. Orderings on general noncommutative rings have received less attention. Inwx 13, Chap. 6 , Lam proves basic properties and gives some history; also seewx 7 . In w 24 x , Powers introduces the real spectrum of higher level of a noncommutative ring. In the present paper we show that orderingsŽ. of level 1 on noncommutative rings give rise to abstract real spectra exactly as in the commutative case. As part of the proof, we show that if ᒍ is a real prime in a noncommuta- tive ring A, then the orderings on A having support ᒍ form a space of orderings in a natural way. We show that if ᒍ is a real prime of A, then ab g ᒍ « a g ᒍ or b g ᒍ Ž.so Arᒍ is an integral domain . Orderings on A with support ᒍ corre- spond to orderings on Arᒍ with supportÄ4 0 . In the commutative case, supportÄ4 0 orderings on an integral domain correspond to orderings on the field of fractions. In the noncommutative case, the situation is more complicated since there are integral domains having supportÄ4 0 orderings which cannot be embedded in a skew field. It is clear that additional work remains to be done. For example, real places on noncommutative integral domains need to be examined and the connection of these with support zero orderings should be looked at. Also, one should study specialization of orders and the noncommutative ana- logue of the real holomorphy ring. These topics will be dealt with in a later paper. Of course one would also like to find some geometric application of the noncommutative real spectrum, but it is not clear how to proceed with this.

1. INTRODUCTION

Throughout, A denotes aŽ. not necessarily commutative ring with 1. Orderings on A are defined as in the commutative case. 414 LEUNG, MARSHALL, AND ZHANG

DEFINITION 1.1. A subset P : A is said to be an ordering of A if P q P : P, PP : P, P jyPsA, and P lyP is a prime ideal of A. The prime ideal P lyP is called the support of the ordering P. A prime ideal of A is said to be real if it is the support of some ordering of A. Our first result is basic and perhaps a bit surprising although the proof is simple enough.

THEOREM 1.2. If ᒍ : A is a real prime then Arᒍ is an integral domain, i.e., if ab g ᒍ then a g ᒍ or b g ᒍ. Proof. Fix an ordering P with P lyPsᒍ. Working with the in- duced ordering Prᒍ on the factor ring Arᒍ, we are reduced to the case where P lyPsᒍsÄ40. 2 2 Claim.Ifab s 0, then either a s 0orb s0. Clearly we can assume 2 a,bgP. Then either a y b g P or b y a g P, so either Ž.a y bbsyb 22 2 gPor abŽ.ya sya gP. Thus either b g P lyPsÄ40oragP lyPsÄ40. 2 By the Claim, it remains to show that a s 0 « a s 0. Since the ideal Ä40 is prime, it suffices to show that axa s 0 for all x g A. Clearly we can assume x g P. Either xa y ax g P or ax y xa g P. Consequently, either Ž.xa y ax a syaxa g P or aax Ž.yxa syaxa g P so, in either case, axa g P lyPsÄ40.

DEFINITION 1.3. We define the real spectrum Sper A as in the commu- tative case. As a set Sper A is just the set of all orderings of A. The sets Ä4PgSper A ¬ a f yP , a g A, form a subbasis for the topology on Sper A. For the definition of a spectral space, seewx 2 or w 21 x , for example.

THEOREM 1.4. Sper A is a spectral space. If P, P12, P g Sper A, P : Pi, is1, 2 then P12: PorP 21:P. In view of Theorem 1.2, Theorem 1.4 can be proved exactly as in the commutative case; seewxw 2, Remark 7.1.17 and Proposition 7.1.22 or 12, Propositions 4.2 and 4.8x . Alternatively, it is a consequence of Theorem 1.5Ž. 2 below usingwx 21, Propositions 6.3.3 and 6.4.1 . We need some notation. For a g A, let a: Sper A ª Ä4y1, 0, 1 be defined by

¡1ifagP_yP aPŽ.s~0ifagPlyP ¢y1ifagyP_P THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 415

Ä4 and let GA s a ¬ a g A . For a real prime ᒍ : A, let Sperᒍ A denote the set of orderings of A having support ᒍ and let GAᒍ denote the set of restrictions Ä4a ¬¬aAᒍ. Sper ᒍ A g_ THEOREM 1.5. Suppose Sper A / л. Then:

Ž.1 For each real prime ᒍ of A, the pair ŽSperᒍᒍA, G A. is a space of orderings. Ž.2 The pair ŽSper A, GA . is an abstract real spectrumŽ i.e., a space of signs.. In Sections 5 and 6 we prove Theorem 1.5. Also, descriptions of value sets and transversal value sets are given; see Theorems 5.2, 6.1 and 6.2. These are the main goals of the paper. The reader should refer towx 1, 21 for the general theory of spaces of orderings and abstract real-spectra. The most striking consequences of this theory in the present context are various local-global principles relating properties ofŽ. Sper A, GA with corresponding properties of the residual spacesŽ. SperᒍᒍA, GA,ᒍa real prime. In the commutative case, orderings on A with support ᒍ correspond naturally to orderings on the field of fractions ␬Ž.ᒍ of Arᒍ, and Ž.SperᒍᒍA, GAis naturally identified with the space of orderings of ␬ Ž.ᒍ . In contrast, the situation in the noncommutative case is more complicated, as the following example shows. EXAMPLE 1.6. Inwx 14 Malcev gives an example of an integral domain which cannot be embedded in a skew field. Inwx 5, 27 Chehata and Vinogradov prove, simultaneously and independently, that the example given by Malcev can be ordered. Consequently, noncommutative integral domains exist which have supportÄ4 0 orderings but cannot be embedded in a skew field. Malcev’s example is a semigroup ring ޚŽ.T . T is the factor semigroup T s Sr; where S is the free semigroup on the letters a, b, c, d, x, y, u, ¨ and ; is generated by ax ; by, cx ; dy, and au ; b¨. Malcev proves that T is a cancellation semigroup which is not embeddable in a group and that ޚŽ.T is an integral domain which is not embeddable in a skew field. Chehata and Vinogradov show that T possesses a total ordering respecting the multiplication.Ž In fact there are many such order- ings.. For any such ordering - on T we have an associated supportÄ4 0 Ž. ordering P : ޚ T consisting of all elements mt11qиии qmtnnwith m ig ޚ,ti gT,t1 -иии - tn, and m1 ) 0 together with the element 0.

2. PREORDERINGS AND ORDERINGS 2 Notation 2.1. For any subset S : A, ASŽ.will denote the set of all permuted products of elements a11, a ,...,ann,a ,s1,...,sm, for a1,...,an 416 LEUNG, MARSHALL, AND ZHANG

2 2 g A, s1,...,smgS,nG0, m G 0. For example, each of a112ss,a 121ss, 2 2 2 asas1112, assa 112 1, asas 12 11, assa 1211, sa 112s, sasa 112 1, ssa 12 1, sa 2 11s, 22 sasa2111,ssa 211is a permuted product of a1, a 1, s 1, s 2. ⌺ ASŽ.will denote the set of all finite sums of elements of AS2Ž..

2 Observe: For any ordering P of A, ⌺ APŽ.sP. Consider any per- muted product of a11, a ,...,ann,a ,s1,...,smfor a1,...,angA, s1,...,sm gP. Using the fact that P jyPsA together with the fact that y1 commmutes with elements of A, we can assume a1,...,an gP,so PP : P implies the permuted product is in P.

DEFINITION 2.2. A preordering of A is a subset T : A such that 2 ⌺ATŽ.sT. A preordering T of A is said to be proper if y1 f T. Every ordering is a proper preordering. For any subset S of A, ⌺ AS2 Ž. is the smallest preordering of A containing S. In particular, ⌺ A2ŽÄ41is. the unique smallest preordering of A. In the rest of the paper, we denote ⌺A2 ŽÄ41by. ⌺A22for short. If A is commutative, then ⌺ A consists of all sums of squares and T is a preordering if and only if T q T : T, TT : T, 2 and ⌺ A : T. Notation 2.3. For a preordering T of A and any a g A, we denote by Tawxthe set of all finite sums of permuted products of a11, a ,...,ann,a ,t1, ...,tm,a, for a1,...,an gA, t1,...,tm gT, nG0, m G 0. Clearly T q Tawxis a preordering and it is the smallest preordering containing T j Ä4a .If Ais commutative then TawxsTa. Inwx 13 , an m-system is defined to be a non-empty set s : A such that a, b g S « there exists c g A such that acb g S. The compliment of a prime ideal is an m-system.

DEFINITION 2.4. A non-empty set S : A is called a quadratic m-system 2 if for any a, b g S, there exists c g A such that ac b g S. The motivation behind this definition comes from the next lemma.

LEMMA 2.5. If ᒍ is a prime,2fᒍ,and a, b g A_ᒍ, then there exists 2 c g A such that ac b f ᒍ. Proof. Note: if d f ᒍ, then 2 d f ᒍ. ᒍ is prime so d, e f ᒍ « there exists x g A such that dxe f ᒍ. Taking e s 2 and using dx2 s Ž.2 dx yields 2 d f ᒍ as required. Now let y g A be such that ayb f ᒍ. By the above note, 4ayb s 22Ž.ayb f ᒍ. But

22 4ayb s ayŽ.q1 byay Ž.y1 b

22 so either ayŽ.q1 bfᒍor ay Ž.y1 bfᒍ. THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 417

THEOREM 2.6.Ž. 1 Suppose S : T : Q : A where S is a quadratic m-sys- tem, T is a preordering, and Q is a T-moduleŽ i.e., Q q Q : Q and sgQ«Tswx:Q. such that yS l Q s л. If Q is maximal with these properties then Q lyQ is a prime ideal and Q jyQsAiŽ .e., Qisa semi-ordering of A.. Ž.2 Suppose S : T : A where S is a quadratic m-system and T is a preordering such that yS l T s л. If T is maximal with these properties, then T is an ordering. X Proof. Ž.1 First we show ᒍ s Q lyQ is an ideal. Let Q s Äb ¬ 2b g XX Q4. Then Q = Q is an T-module. If ys g Q for some s g S, then X y2 s g Q so ys sy2sqsgQ, a contradiction. Therefore, yS l Q s X л. By maximality of Q, Q s Q . Clearly, ᒍ q ᒍ : ᒍ and yᒍ s ᒍ. Given 2 2 X agA,bgᒍ, we have 4ab s Ž.a q 1 b y Ž.a y 1 b g Q. Since Q s Q, this implies ab g Q. Similarly, yab g Q,soab g ᒍ. A similar argument shows that ba g ᒍ. Next we show ᒍ is a prime ideal. Suppose aAb : ᒍ, a, b f ᒍ. We may assume a, b f Q. Then Q q Tawxand Q q Tb wxare T-modules which contain Q properly so, by maximality of Q, we have equations ys11s q q t1222,yssqqt where s 12, s g S, q 12, q g Q, and t 1g Tawx,t 2gTb wx. 2 2 Since S is a quadratic m-system, there exist c, d g A such that sc11sds 2 gS. The following identity holds:

2 2 2 2 2 2 ysc112111121112sdssŽ.Ž.sqtcsqtdsysc Ž.s qtds 2 2 2 2 2 2 yŽ.s11qtcsd 1211sytctdŽ.s 22qt qtc 112tdt.

Ž.2Ž.2 Ž. Clearly, s11q tcs 11qtds 2gQ. Since s12, s g T, y s 11q t s q 1g Ž. 2Ž.2 Ž Qand y s22q t s q 2g Q, it follows that ysc111 s qtds 2,ys 1q .22 22Ž. tcsds112, and ytct 1122 d s qt are in Q. Finally, since aAb : ᒍ,itis 2 2 2 2 clear that tc11tdt 2gQ. This gives ysc11sds 2gQ, a contradiction. If a, ya f Q, then using the same argument as above, but with b sya, we arrive at a contradiction in exactly the same way, although the justifica- 22 Ž 22 tion of the statement tct11 dt 2gQis a bit different. Since yct1 dt 2 22 . gTand yt11s s q q 1g Q, we have tct112 dt gQ. This proves Q j yQsA. Ž.2 Let Q : A be a T-module containing T and maximal such that ySlQsл. Then Q s T. For otherwise, if a g Q_T, then T q Tawxis a preordering containing T properly, and yS l ŽT q Tawx.sлsince TqTawx:Q. Thus, byŽ. 1 , T is an ordering. COROLLARY 2.7. Any proper preordering of A is contained in an ordering 2 of A. Consequently, A has an ordering if and only if y1 f ⌺ A . 418 LEUNG, MARSHALL, AND ZHANG

Proof. Take S s Ä41 in Theorem 2.6Ž. 2 . Theorem 2.6 generalizes well-known results in the commutative case, e.g., seewx 3, 22, 28 for Theorem 2.6Ž. 1 andwx 2, 12, 21 for Theorem 2.6Ž. 2 . In 1 the commutative case, by going to the localization A ª Sy A, it suffices to deal with the case where S s Ä41 but, in the noncommutative case, localization is not well-behaved. Seewx 24, Proposition 1.9 for a noncommu- tative higher level version of Corollary 2.7.

3. T-COMPATIBLE PRIMES

DEFINITION 3.1. A prime ideal ᒍ of A is said to be T-compatible Ž.where T is a preordering of A if there exists an ordering P = T with ᒍ s P lyP. Clearly a T-compatible prime ideal is real. Also, a prime ideal is real iff it is ⌺ A2-compatible.

THEOREM 3.2. Suppose ᒍ is a prime ideal of A and T is a preordering of A with T lyTsᒍ. If T is maximal with this property, then T is an ordering. Proof. 2 f ᒍ.If2gᒍ, then y1 s 1 qyŽ.2gT,so1gᒍ, a contra- diction. Let S s T _ᒍ. It is clear that S : T, yS l T s л, and, by Lemma 2.5, S is a quadratic m-system. Let P = T be a preordering maximal such that yS l P s л. By Theorem 2.6Ž. 2 , P is an ordering. If P lyPsᒍ we are done. Suppose, to the contrary, that there exists 2 a g P lyP, afᒍ. By Lemma 2.5, there is c g A such that ac a f ᒍ. 2 Then yac a gySlP, a contradiction. As in the commutative case, we have a characterization of T-compatible primes:

COROLLARY 3.3. For T a preordering of A and ᒍ a prime ideal of A, the following are equi¨alent: Ž.1 ᒍis T-compatible. Ž.2 s,tgT and s q t g ᒍ « s, t g ᒍ. Ž.Ž.Ž.3Tqᒍly Tqᒍ sᒍ. Proof. Ž.1 « Ž.2 . Let P be an ordering of A such that P = T and P lyPsᒍ. Suppose s q t g ᒍ, s, t g T. Then s g P and s gytqᒍ :yP,so sgPlyPsᒍ. A similar argument shows that t g ᒍ. Ž.Ž. Ž.Ž. 2«3 . It suffices to show T q ᒍ ly Tqᒍ :ᒍ. Suppose a s t1 Ž. qx1221212sy t qx , t ,t gT, x , x gᒍ. Then t 1212q t syx yx gᒍ Ž. so 2 implies t12, t g ᒍ. Hence a g ᒍ. Ž.3« Ž.1 . Apply Theorem 3.2 to the preordering T q ᒍ. THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 419

Of course, Corollary 3.3 also provides a characterization of real primes:

COROLLARY 3.4. For a prime ᒍ in A, the following are equi¨alent: Ž.1 ᒍis real. 2 Ž.2 s,tg⌺A and s q t g ᒍ « s, t g ᒍ. 2 2 Ž.3 Ž⌺Aqᒍ.Žly ⌺A qᒍ.sᒍ. Also, using Theorem 3.2, one can derive the following result which is essentially due to Lamwx 13, Lemma 17.8 . THEOREM 3.5. Let T be a preordering of A such that ᒍ [ T lyTisa prime ideal, and let a g A. Then the following are equi¨alent: Ž.1 afP for all orderings P of A with T : P and P lyPsᒍ. Ž.2 ŽTqTawx.Žly TqTawx.pᒍ. Ž.3 ysa s t for some s, t g T _ᒍ. Proof. Ž.1 « Ž.2.If ŽTqTawx.Žly TqTawx.sᒍ, then we can apply Theorem 3.2 to get an ordering P with P = T q Tawxand P lyPsᒍ. This contradictsŽ. 1 . Ž.2« Ž.3.TlyTsᒍ implies, in particular, that ᒍ is T-compatible. Ž. Suppose t11q s sy t 22qs fᒍ with t12, t g T, s 12, s g Tawx. Then Ž. ysa s t where s s t12q t , t s s 12q sa. Since ᒍ is T-compatible and since t12, t , sa 1,sa 2 are elements of T and a f ᒍ, it is clear that s, t f ᒍ. Ž.3« Ž.1.IfagPfor some ordering P with P = T and P lyPsᒍ, then t sysa g P lyPsᒍ, a contradiction.

4. THE POSITIVSTELLENSATZ

As in the commutative casewx 2, 12 , various versions of the Positivstellen- satz hold. We mention some of these now, in passing. We continue with the notation of Section 1. We assume T is a preordering of A and XsÄ PgSper A ¬ tPŽ.G0 for all t g T4. The versions of the Positivstel- lensatz that we present hereŽ. see Corollary 4.2 can all be derived from Theorem 2.6 via the following result. In fact, Theorem 2.6 itself can be viewed as some sort of abstract Positivstellensatz.

THEOREM 4.1. Suppose c, d g A. Then c G 0 « d s 0 holds on X iff 2 k 2 yd gTq⌺Awx c for some integer k G 0. 2 k 22 Proof. Let S s Äd ¬ k G 04 and consider the ⌺ A -module T q ⌺ Acwx. 2 2 If yS l ŽT q ⌺ Acwx.sлthen there is a ⌺ A -module Q containing 2 Tq⌺Acwxwhich is maximal subject to the condition yS l Q s л.By Theorem 2.6Ž. 1 , ᒍ s Q lyQ is a prime ideal. Also, T : Q,soT is 420 LEUNG, MARSHALL, AND ZHANG

ᒍ-compatible. Let Tᒍ s T q ᒍ. Since d f ᒍ, it follows from our assump- tion that c f P for all orderings P = Tᒍ having support ᒍ, so by Theorem 2 3.5, ysc s t for some s, t g Tᒍᒍ_ᒍ. Then st g T : Q and yst s s c g 2 ⌺Acwx:Q,so st g Q lyQsᒍ, a contradiction. COROLLARY 4.2Ž. Positivstellensatz . Suppose a, b g A. Then 2 Ž.1 as1on X m y1 g T y ⌺ Aawx. 2k Ž.2 as0on X m ya g T for some integer k G 0. 2 k 2 Ž.3 aG0on X m ya g T y ⌺ Awx a for some integer k G 0. 222k 2 Ž.4 asbonXmyŽ.aqb gTy⌺Awx ab for some integer k G 0. Proof. Ž.1 Take c sya, ds1.Ž. 2 Take c s 0, d s a.Ž. 3 Take c s 22 ya,dsa.Ž. 4 Take c syab, d s a q b . Remark 4.3. Occasionally one wants to replace X by Y s Ä P g Sper A ¬tPŽ.G0 for all t g T and sPŽ./0 for all s g S4where T : A is a preordering and S : A is a multiplicative set. It is worth noting that our results can be made to apply in this case also: By compactness, the condition c G 0 « d s 0onY is equivalent to the condition that there exists s g S such that c G 0 « dss 0on X.

5. THE SPACE OF ORDERINGS ATTACHED TO A REAL PRIME

2 We prove a generalization of Theorem 1.5Ž. 1 . Suppose that y1 f ⌺ A 2 and fix a proper preordering T of A. For example, take T s ⌺ A . Also fix a T-compatible prime ᒍ and denote by Xᒍ the set of all orderings of A containing T and having support ᒍ. By definition, Xᒍ / л. For a g A_ᒍ, Ä4 let a: Xᒍ ª y1, 1 be defined by

1ifagP aPŽ.s ½y1ifgyP

Ž.so a is the restriction to Xᒍ of the mapping a defined in Section 1 . Also, Ä4 let Gᒍs a ¬ a g A_ᒍ . By Theorem 1.2, ab s ab for all a, b g Gᒍᒍ,soG Ä4Xᒍ is a subgroup of 1, y1 . Let Tᒍᒍs T q ᒍ. T is a preordering, and TᒍᒍlyT sᒍ by Corollary 3.3. By Theorem 3.5, a s b iff sab s t for ²: Ä4 some s, t g Tᒍ _ᒍ. We define value sets as inwx 21 , i.e., Dasa, for all ²:Ä Ž. Ž.Ž.Ž. Ž. Ž. agGᒍᒍ,Da,bscgG¬cP qaPbPcP saP qbP for all THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 421

PX4, for all a, b G , and Da²:,...,a Da ²:,c, gᒍᒍg 1ncsD gD²a2,..., an: 1 Ž. for all a1,...,angGᒍᒍ,if nG3. We want to show that the pair X , Gᒍis a space of orderings as defined inwx 20, 21 , i.e., that the following axioms hold:

Ž. Ä4Xᒍ AX1 Xᒍᒍ/ л, G is a subgroup of y1, 1 , Gᒍcontains the con- stant function y1, and Gᒍᒍseparates points in X . Ž. Ä4 Ž. Ž.Ž.Ž . AX2 If f: Gᒍ ª y1, 1 satisfies fabsfafb, fy1sy1, ²:Ž.Ž. Ž. and c g Da,b,fasfbs1«fcs1, then there exists P g Xᒍ Ž. Ž . such that fasaP for all a g Gᒍ . Ž. ²: ² : AX3 For all a123, a , a g Gᒍ,if bgDa123,c for some c g Da,a , ²: ² : then b g Dd,a31for some d g Da,a2. The proof is quite similar to the proof in the field casewx 21 or in the skew field casewx 6, 20, 26 . On the other hand, in view of Malcev’s example Ž.see Example 1.6 it is not possible to deduce this result directly from the corresponding result for skew fields. The main step in the proof is in the following lemma: ²:Ä 4 LEMMA 5.1. Da,bscgGᒍᒍ¬cgTawxqTb ᒍ wx,cfᒍ. Proof. By going to the factor ring Arᒍ, there is no harm in assuming Ä4 ᒍs0 . The inclusion = is clear. In fact, if c g TaᒍᒍwxqTb wx,cfᒍ, then, for P g Xᒍ , a, b g P « c g P, and a, b gyP«cgyP. This Ž.Ž. Ž.Ž. means either cPaP s1orcPbP s1 for all P g Xᒍ,socg Da²:,b. To show the other inclusion, let c g Da²,b :. Then either aPcPŽ.Ž.s1 Ž.Ž. XYX or bPcP s1 for all P g Xᒍᒍ. Let T s T q Tᒍwxyac and T s T q X Ž. Ž. Twxybc . Since either ac P s 1orbc P s 1 for all P g Xᒍ , we have either ac g P or bc g P for all P g Xᒍ . Thus there is no ordering with YYY supportÄ4 0 containing T so, by Theorem 3.2, T lyT /Ä40 . There are two cases:

XX Case 1. T lyT sÄ40 . Then by Theorem 3.5, tbc s s for some X 2 2 s, t g T , s, t / 0, so t bc s ts / 0. Let t bc s t11y s where t 1g Tᒍ, 22 s1 gTacᒍwx.If t11/0 then tcstbcqsc 1/0. Then c s tc 1and 22 sc1 gTaᒍᒍwxand tbcgTbwxso tc1gTaᒍᒍ wxqTb wx.If t1s0 then y s 22 Ž.2 Ž.2 tbcsysc1 /0 and y g TbwxlyTa wx. Then 2 ycsyycq1 y Ž.2 2 yycy1 gTaᒍᒍwxqTb wxandŽ. 2 ycsc since y / 0, 2 / 0. XX Case 2. T lyT /Ä40 . Then, again by Theorem 3.5, we have tac s s 2 for some s, t g Tᒍᒍ, s, t / 0. Then c s sc and sc s tac g Tawx. Using Lemma 5.1, we can now prove the desired result. 422 LEUNG, MARSHALL, AND ZHANG

THEOREM 5.2. Suppose T is a proper preordering in A and ᒍ is a T-compatible prime, notations as abo¨e. Then

Ž.1 ŽXᒍᒍ,G . is a space of orderings. Ž. ² : Ä 4 2 Da1,...,an s b¬bgTaᒍwx1qиии qTaᒍ wxn,bfᒍ. Ž. ² : Ä XX 3 Da1,...,an s b¬bgA_ᒍ and there exist a1,...,an gA_ᒍ XXX 4 such that b s a1 q иии qanii and a s a for i s 1,...,n. Ž. Ž . Ä4 Proof. 1 Axiom AX1 is clear. Let f: Gᒍ ª y1, 1 satisfy the conditions ofŽ. AX2 and let P s Äa g A_ᒍ ¬ faŽ.s14jᒍ. Since fŽ.1 s 1, we see that T : P. Also, it is easy to check that P jyPsA,PlyP sᒍ,PP : P, and P q P : P.Ž To check that P q P : P, use the fact that ²: . Ž. aqbgDa,b if a q b f ᒍ. This means P g Xᒍ and clearly fas Ž. Ž. aP for all a g Gᒍ . Thus AX2 is satisfied. Ž. For AX3 , by Lemma 5.1, c s c11for some c g Taᒍwx2qTaᒍ wx3. Then ²: bgDa11,c so, again by Lemma 5.1, b s b1for some b1g Taᒍwx1q Tcᒍwx1:Taᒍ wx1qTaᒍ w2 xqTaᒍ wx31123. Let b s␣ q ␣ q ␣ , ␣ig Taᒍwxi,is 1, 2, 3. If ␣12q ␣ f ᒍ, take d s ␣12q ␣ .If ␣ 12q␣gᒍ, then b ²: sb133s␣saso we can take d to be any element in Da12,a in this case. Ž.2 This follows from Theorem 3.5 and Lemma 5.1. Suppose n s 1. Clearly if b g Taᒍwx,bfᒍ, then b s a. Conversely, if b s a then abs 1 2 so by Theorem 3.5, sab s t for some s, t g Tᒍᒍ_ᒍ. Then tb s sab g Tawx and b s tb. For n s 2 the result is immediate from Lemma 5.1. Suppose nG3. Suppose b s s121q s , s g Taᒍwx12,sgTaᒍ wx2qиии qTaᒍ wxn,bfᒍ.If ²: ² : s21fᒍthen b g Da,s1and by induction on n, s2g Da2,...,an,so ²: ²: bgDa1,...,an .If s21gᒍthen s f ᒍ and b s s 1s a11g Da,...,an. ²: ²:² Conversely, suppose b g Da1,...,an . Then b g Da12,c,cgDa,..., : an . By induction on n, c s c11, c g Taᒍwx2qиии qTaᒍ wxn, and b s b1, b1gTaᒍwx1qTcᒍ wx1:Taᒍ wx1qиии qTaᒍ wxn. Ž. XX X X 3Ifbsa1qиии qanii, b f ᒍ, a s a , i s 1,...,n, then b g Taᒍwx1 X ²:Ž. qиии qTaᒍwxn so b g Da1,...,an by 2 . For the inclusion : , suppose ²: bgDa1,...,an . Let a0 syb. From standard properties of values sets Ž.for example, see the description of value sets given in Remark 5.3 below ²: we know that y aig Da0,...,aiy1,aiq1,...,an holds for i s 0,...,n. Ž. X Using this and 2 this gives us n q 1 equations, ybijs Ý /iijt , i s XXXX 0,...,n with biiijs a , t g Taᒍwxjiij. Thus, if a s b q Ý /ijit , then a is a i XX X and ya01s a q иии qan.

Remark 5.3. Various results now follow automatically from the theory of spaces of orderings. For the reader unfamiliar with spaces of orderings THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 423 we point out some of these results: Ž. ² 1 We have yet another characterization of value sets: b11g Da, :ŽÝnn.Ý Ž. ...,an iff there exists b2,...,bngGᒍ such that is1aPiis s1bPi holds for all P g Xᒍ wx21, p. 26 .

Ž.2 One can form the Witt ring WXŽ.ᒍᒍ,G , and the representation Ž. theorem holds: A continuous function ␾: Xᒍᒍª ޚ belongs to WX,Gᒍ Ý ␾Ž. << iff P g V P ' 0 mod V holds for all finite fans V : Xᒍ wx21, p. 41 .

Ž.3 ŽXᒍᒍ,G .possesses a P-structurewx 21, p. 79 . In the paper which is the sequel to this we show more is true, namely Ž.Xᒍᒍ, G has a natural P-structure coming from real places on the integral domain Arᒍ and also Brocker’s¨ trivialization theorem for fans holds. Ž.4 The isotropy theorem holds. Also, the structure theorem for spaces of orders of finite chain length applies to Ž.Ž.Xᒍᒍ, G if X ᒍᒍ, G has finite chain lengthwx 21, pp. 65 and 67 .

6. THE MAIN THEOREM

In the previous section we considered orderings on A having as support a fixed real prime. In this section we look at the complete set of orderings of A. As in Section 1, if a g A, a: Sper A ª Ä4y1, 0, 1 is defined by

¡1ifagP_yP aPŽ.s~0ifagPlyP ¢y1ifagyP_P and GA s Ä4a ¬ a g A . We want to prove Theorem 1.5Ž. 2 . Also, we want to describe the value sets Da²:1,...,an and the transversal value sets t Da²:1,...,an in this situation. We prove a more general result. Namely, we fix a preordering T in Awith y1 f T, and we let X s Ä P g Sper A ¬ tPŽ.G0 for all t g T4. Also, we denote by G the set of all restrictions of elements of GA to X. Value sets and transversal value sets are defined as inw 21, pp. 99 and t²: Ä4 t²:Ä Ž. Ž. 105x , i.e., Dasa,Da12,a scgG¬aP1qaP 2 s Ž. Ž.Ž. Ž. 4 t²: cP qcPa12 Pa P,forall PgX,and Da1,...,ans t t Da²:,c,if n 3, for all a, a ,...,a G, and Dcg D ² a2,...,an: 11G ng ²: Ä 2 4 ²:Ä 2Ž.Ž. 2Ž. Ž. Dasba¬bgG, Da12,a scgG¬cPaP1 qcPaP2 s Ž. Ž. Ž.Ž. 4 ²: cP qaPaPcP12 ,forall PgX,and Da1,...,ans Da²:,c,if n 3, for all a, a ,...,a G. We want to Dcg D² a2,...,an: 11G ng show the pair Ž.X, G is an abstract real spectrum, i.e., that it satisfies the 424 LEUNG, MARSHALL, AND ZHANG following axioms: X Ž.AX1 X / л, G is a submonoid of Ä4y1, 0, 1 , G contains the con- stant functions y1, 0, 1, and G separates points in X. Ž.AX2 If S is a submonoid of G satisfying S jySsG,y1fS, t a,bgS«Da²:,b:S, and ab g S lyS«agSlyS or b g SlyS, then there exists P g X such that S s Äa g G ¬ aPŽ.G0.4 Ž. t²: t²: AX3 For all a123, a , a g G,if bgDa1,cfor some c g Da23,a, t²: t²: then b g Dd,a31for some d g Da,a2. THEOREM 6.1. Suppose X s ÄP g Sper A ¬ tPŽ.G0for all t g T4 where T is a proper preordering of A, and let G be the set of restrictions of elements of GA to X. Then Ž.1 ŽX,G . is an abstract real spectrum. Ž. ² : Ä 2 For any a1,...,an gA, Da1,...,an s b¬bgTawx1 qиии q Tawxn4. Ž. t² : Ä XX 3 For any a1,...,an gA, Da1,...,an s b¬there exists a1,...,an XXX 4 gA with b s a1 q иии qanii and a s a , i s 1,...,n. Here, in Ž.2 and Ž.3 it has to be understood that a denotes the restriction of a to X. Theorem 6.1 generalizes exactly known results in the commutative case, e.g., seewx 21, Propositions 5.5.1, 5.5.4, and Theorem 6.1.2 . We also have the following generalization of Theorem 6.1 which, in the commutative case, 1 can be deduced from Theorem 6.1 by going to the localization A ª Sy A. Of course, in the noncommutative case, localization is not well behaved. COROLLARY 6.2. Suppose Y s Ä P g Sper A ¬ tPŽ.G0for all t g T and sPŽ./0for all s g S4/ л, where T : A is a preordering and S : Aisa multiplicati¨e set and let H be the set of restrictions of elements of GA to Y. Then Ž.1 ŽY,H . is an abstract real spectrum. Ž.2 Value sets and trans¨ersal sets forŽ. Y, H are described exactly as in Theorem 6.1Ž. 2 and Ž.3 but with the understanding that now a denotes the restriction of a to Y. Proof. In view of Theorem 6.1, this follows fromwx 21, Proposition 6.5.7 .

Proof of Theorem 6.1. AxiomŽ. AX1 is clear. Let S be a submonoid of Gsatisfying the conditions ofŽ. AX2 and let P s Ä4a g A ¬ a g S . Then one checks that P g X and S s Äa g G ¬ aPŽ.G04 . The main point is t that a q bg Da²:,bso P q P : P. This provesŽ. AX2 . THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 425

Thus, to proveŽ. 1 , we are left with showing Ž AX3 . . According tow 21, Theorem 6.2.4x ,Ž. AX3 is equivalent to a certain weak version of Ž. AX3 Ž.called weak associativity ,

²: ² : ² : if b g Da12,c, for some c g Da,a33, then b g Dd,a ²: for some d g Da12,a ,

t provided the condition Da²:,b/лholds for all a, b g A ŽŽ.i.e., AX3a t andŽ. AX3b inwx 21.²: . Since a q bg Da,b, the latter condition is clear, and weak associativity is more or less immediate once we have the description of value sets given inŽ. 2 . So we turn our attention toŽ. 2 and Ž. 3 . For each of these, the inclusion = is easy, so we concentrate on proving the other inclusion. Also, from the description of transversal value sets given inwx 21, Proposition 6.2.7 , one sees that it is possible to deduceŽ. 3 from Ž. 2 . According tow 21, Proposition t²: ² : 6.2.7x , y a01g Da,...,aniholds iff y a g Da0,...,aiy1,aiq1,...,an holds for i s 0,...,n. Using this andŽ. 2 this gives us n q 1 equations, XXXX bijsÝ/iijt,is0,...,n with b is a i, t ijg Tawx j. Thus, if a is b iq Ý j/ijit , XXXX then aiis a and ya01s a q иии qan. Thus we are left with proving the non-trivial inclusion : ofŽ. 2 . As in the commutative case, this reduces to the case n s 2, but the proof for ns2 given in the commutative caseŽ seewx 21, Proposition 5.5.1. does not seem to generalize easily. Here we take a different approach which has elements in common with the approach taken inwx 1 ; namely we use Theorem 2.6Ž. 1 . Ä Suppose a g A is such that a is not in the set b ¬ b g Tawx1 4 2kq1 qиии qTawxn . Then, for each k G 0, a is not in the T-module yTawx 2kq1 qTawx1 qиии qTa wxn. For if a sysqs1qиии qsn with s g Tawx 2kq1 kq1 and siig Tawx, then a q s s s1q иии qsnand a s a2q s which 2k contradicts our assumption. This implies that ya f T q Twxyaa1 qиии qTwxyaan for k G 0. Let Q be a T-module containing T q Twxyaa1 qиии qTwxyaan and maximal subject to the condition yS l Q s л where Ä 2 k 4 S s a , k G 0 , and let ᒍ s Q lyQ,Tᒍ sTqᒍ. Then ᒍ is T-compat- Ž . ible and Tᒍᒍl Taawx1qиии qTaaᒍ wxn:QlyQsᒍ. Since ᒍ is a real Ž . prime and a f ᒍ, this implies that TaᒍᒍwxlTawx1qиии qTaᒍ wxn:ᒍ. Let Ä4 Ž. Xᒍ sPgX¬PlyPsᒍ and consider the residue space Xᒍᒍ, G . Ä4 Here Gᒍᒍs b ¬ b g A where bᒍdenotes the restriction of b to Xᒍ. U According to Theorem 5.2, the pair ŽXᒍᒍ, G .is a space of orderings where U Ä4 GᒍᒍᒍsG_0. ²: ²: Claim. aᒍ f Da1ᒍ,...,anᒍᒍ. For otherwise, since Da1 ,...,anᒍs Ä4 bᒍᒍ¬bgTawx1qиии qTaᒍ wxnby Theorem 5.2, we would obtain a s b on 426 LEUNG, MARSHALL, AND ZHANG

Ž . Xᒍᒍfor some b g Tawx1qиии qTaᒍ wxn_ᒍ. By Theorem 3.5, there exist 22Ž s,tgTᒍ_ᒍwith sab s t. Then sab s tb so sab g TaᒍᒍwxlTawx1 . qиии qTaᒍwxn :ᒍ, which is a contradiction since a, s, b f ᒍ.

Ž. From the Claim using the obvious fact that Xᒍ : X , it follows that ²: afDa1,...,ani. Note that some of the a may be in ᒍ, but this does not alter the validity of the argument. This completes the proof of Theo- rem 6.1.

REFERENCES

1. C. Andradas, L. Brocker,¨ and J. Ruiz, Constructible sets in real geometry, in Ergeb. Math. Grenzgeb., Vol. 33, Springer-Verlag, New YorkrBerlin, 1996. 2. J. Bochnak, M. Coste, and M.-F. Roy, Geometrie´´ Algebrique ´ Reelle, ´ in Ergeb. Math. Grenzgeb., Springer-Verlag, New York, 1987. 3. L. Brocker,¨ Positivbereiche in kommutativen Ringen, Abh. Math. Sem. Uni¨. Hamburg 52 Ž.1982 , 170᎐178. 4. L. Brocker,¨´´Abstract real spectra, preprint, Seminaire de Structures Algebriques Or- donnees,´ Paris 7, 1993. 5. C. G. Chehata, On an ordered semigroup, J. London Math. Soc. 28 Ž.1953 , 353᎐356. 6. T. Craven, Witt rings and orderings on skew fields, J. Algebra 77 Ž.1982 , 74᎐96. 7. L. Fuchs, Partially ordered algebraic systems, in International Series of Mongraphs on Pure and Applied Mathematics, Vol. 28, Pergamon, Elmsford, New York, 1963. 8. F. Kalhoff, Spaces of orderings and Witt rings of planar ternary rings, J. Pure Appl. Algebra 58 Ž.1989 , 169᎐180. 9. F. Kalhogg, Orderings, algebras and projective planes, Exposition. Math. 13 Ž.1995 , 3᎐38. 10. J. Kleinstein and A. Rosenberg, Succinct and representational Witt rings, Pacific J. Math. 86 Ž.1980 , 99᎐137. 11. M. Knebusch, On the local theory of signatures and reduced quadratic forms, Abh. Math. Sem. Uni¨. Hamburg 51 Ž.1981 , 149᎐195. 12. T.-Y. Lam, An introduction to real algebra, Rocky Mountain J. Math. 14 Ž.1984 , 767᎐814. 13. T.-Y. Lam, A first course in noncommutative rings, in Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, New YorkrBerlin, 1991. 14. A. Malcev, On the immersion of an algebraic ring into a field, Math. Ann. 113 Ž.1937 , 686᎐691. 15. M. Marshall, Classification of finite spaces of orderings, Canad. J. Math. 31 Ž.1979 , 320᎐330. 16. M. Marshall, Quotients and inverse limits of spaces of orderings, Canad. J. Math. 31 Ž.1979 , 604᎐616. 17. M. Marshall, The Witt ring of a space of orderings, Trans. Amer. Math. Soc. 258 Ž.1980 , 505᎐521. 18. M. Marshall, Spaces of orderings, IV, Canad. J. Math. 32 Ž.1980 , 603᎐627. 19. M. Marshall, Spaces of orderings: Systems of quadratic forms, local structure, and saturation, Comm. Algebra 12 Ž.1984 723᎐743. THE REAL SPECTRUM OF A NONCOMMUTATIVE RING 427

20. M. Marshall, A simple system of axioms for spaces of signatures, J. Pure Appl. Algebra 57 Ž.1989 159᎐164. 21. M. Marshall, Spaces of orderings and abstract real spectra, in Lecture Notes in Mathe- matics, Vol. 1636, Springer-Verlag, New YorkrBerlin, 1996. 22. M. Marshall and L. Walter, Signatures of higher level on rings with many units, Math. Z. 204 Ž.1990 , 129᎐143. 23. V. Powers, Higher level reduced Witt rings of skew fields, Math. Z. 198 Ž.1988 , 545᎐554. 24. V. Powers, Higher level orders on noncommutative rings, J. Pure Appl. Algebra 67 Ž.1990 , 285᎐298. 25. V. Powers, Holomorphy rings and higher level orders on skew fields, J. Algebra 136 Ž.1991 , 51᎐59. 26. A. Tschimmel, ‘‘Uber¨ Anordnungsraume¨¨ von Schiefkorpern,’’ Ph.D. dissertation, Univer- sitat¨¨ Munster, 1981. 27. A. Vinogradov, On the theory of ordered semigroups, Sci. Proc. I¨ano¨o Pedagogical Inst. 4Ž.1953 , 19᎐21.wx In Russian 28. L. Walter, ‘‘Quadratic Forms, Orderings, and Quaternion Algebras over Rings with Many Units,’’ Master’s Thesis, Univ. of Sask., 1988.