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The Real Spectrum of a Noncommutative Ring 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 View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector Murray Marshall² and Yufei Zhang³ 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,È 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]. ² 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. ³ E-mail address: [email protected]. 412 0021-8693r97 $25.00 Copyright Q 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 p : G we can form the so-called residue space Xpp, 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. BrockerÈ 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 p is a real prime in a noncommuta- tive ring A, then the orderings on A having support p form a space of orderings in a natural way. We show that if p is a real prime of A, then ab g p « a g p or b g p Ž.so Arp is an integral domain . Orderings on A with support p corre- spond to orderings on Arp 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 p : A is a real prime then Arp is an integral domain, i.e., if ab g p then a g p or b g p. Proof. Fix an ordering P with P lyPsp. Working with the in- duced ordering Prp on the factor ring Arp, we are reduced to the case where P lyPspsÄ40. 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 p : A, let Sperp A denote the set of orderings of A having support p and let GAp denote the set of restrictions Ä4a ¬¬aAp. Sper p A g_ THEOREM 1.5. Suppose Sper A / B. Then: Ž.1 For each real prime p of A, the pair ŽSperppA, 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. SperppA, GA,pa real prime. In the commutative case, orderings on A with support p correspond naturally to orderings on the field of fractions kŽ.p of Arp, and Ž.SperppA, GAis naturally identified with the space of orderings of k Ž.p . 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 ZŽ.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 ZŽ.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 : Z T consisting of all elements mt11q??? qmtnnwith m ig Z,ti gT,t1 -??? - tn, and m1 ) 0 together with the element 0.
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