Real Closed * Rings

Real Closed * Rings Jose Capco October 2010 Doctoral Thesis Submitted to FakultÄatfÄurInformatik und Mathematik UniversitÄatPassau in partial ful¯llment of the requirements for the Degree Dr. rer. nat. Advisor: Prof. Dr. Niels Schwartz Jose Capco PROFACTOR GmbH, Im Stadtgut A2, 407 Steyr-Gleink, Austria Email: 2 Acknowledgements I would like thank my parents, Nicanor and Azar Capco, for supporting me and my endeavour to study. They had to sacri¯ce a lot so that I could travel to Germany and pursue my studies. I would like to thank my ¯anc¶e,Haydee Co, for understanding and supporting my passion for Mathematics. She has always remained by my side during the most di±cult and trying times. I would like to thank my senior, Andreas Fischer, for advising me both on mathematics and on life in the university. I want to extend my gratitude to my friend, Oliver Delzeith, who helped me correct the Preliminaries Section of my Phd Thesis. I have always admired Oliver and wished I had a small fraction of his amazing knowledge in algebraic geometry. I would like to thank my supervisor, Prof. Niels Schwartz, for supporting me and for tolerating, understanding and accepting my shortcomings as his supervisee. Niels was like a father to me, without him this Dissertation would not be possible. I would like to extend my gratitude to the Deutsche Forschungsgemeinschaft (DFG) for their ¯nancial support under the DFG projects SCHW 287/17-1 and SCHW 287/18-2. I would like to thank IVM Technical Consultants and Engineering Center Steyr GmbH & CoKG in Austria for hiring me and making it possible for me to stay in Austria while ¯nishing my PhD in Germany. I would like to thank my current employer, Profactor GmbH, especially the Robotics and Adaptive Systems team. Profactor has allowed me the creativity and freedom that has inspired me in my work. I would like to thank anyone directly and indirectly concerned with my PhD, please accept my apologies if I have failed to include you in this Acknowledgement. Finally, I would like to thank YOU, who are taking the time and patience to read my work. 3 Contents 0 Preliminaries 5 1 Real closed rings and real closed ¤ rings 14 2 The uniqueness of real closure ¤ of Baer regular rings 29 3 The uniqueness of real closure ¤ of regular rings 43 4 Some topologies for f-ring partial orderings of Baer regular rings 58 5 Real closed ¤ reduced partially ordered rings 76 6 Maximal partial orderings and minimal prime spectra of real rings 94 4 x0 Preliminaries Before starting a section, I want to explain the style by which this dissertation is written. Every notation used and de¯ned in a certain section will be inherited by the succeeding section, unless speci¯cally otherwise stated. Unless otherwise stated all the rings in this paper are commutative, associative, unitary and partially ordered rings (porings). Moreover, I very often use the notation "¡ ", which is a short form for "such that". 0 Preliminaries For a smooth and better understanding of the dissertation, a moderate command of topics in Kapitel III of [25] may be necessary. Knowing that many readers not in the ¯eld of real algebra would hopefully also take interest in the work, we attempt to prepare a moderate review of many topics in real algebra and commutative algebra that may otherwise be foreign to the novice reader. For the preliminary section, we assume at least a knowledge in standard abstract algebra and basic algebraic geometry. Otherwise we shall try to make this as self-contained as it possibly and practically can be. So without further ado, let us go about a few de¯nitions and results that will initiate us in real algebra and special topics of algebra. For most of the results in this Section we do not provide a proof, however we cite a reference where complete proofs can be found. All rings, unless otherwise stated, are associative, commutative and unitary. Also, it is to be noted that when I use the term regular ring I mean that the ring is a von Neumann regular ring. We'd like to remind to the reader that in this dissertation everytime I write compact I always mean (like many topologists do) quasicompact and not necessarily Hausdor® . De¯nition. Let A be a commutative unitary ring. We say that a subset A+ ½ A is a partial ordering i® A+ has the following properties : i. a 2 A ) a2 2 A+ i.e. contains all the squares ii. x; y 2 A+ ) x + y 2 A+ i.e. additively closed iii. x; y 2 A+ ) xy 2 A+ i.e. multiplicatively closed iv. x; ¡x 2 A+ , x = 0 i.e. has zero support If there exists a partial ordering A+, we may write that A has a partial ordering A+ or A admits a partial ordering. Associating A with this partial ordering we say that (A; A+) is a partially ordered ring or in short a poring. If A+ is given we may write A and A+ separately rather than as a pair (A; A+). If it is clear, we may also write A is a poring instead of writing (A; A+) is a poring. If A+ satis¯es a ¯fth property that v. x 2 A ) x 2 A+ or ¡x 2 A+ then we call A+ a total ordering of A and we say that A has a total ordering A+. The collection of all porings make up the objects of a category. The category of porings, whose morphisms are just ring homomorphisms (taking 1 to 1) that preserve ordering, i.e. if (A; A+) and (B; B+) are porings, then a morphism between them is represented by f :(A; A+) ! (B; B+) which involves a ring homomorphism (we abuse notation by writing the same symbol for the ring homomorphism) f : A ! B such that f(A+) ½ B+. If it is clear to the reader what the partial orderings A+ and B+ we are dealing with, then we just simply write f : A ! B noting this to be a poring morphism. Notation. Given a ring A we use the following notations AA := fa1a2 : a1; a2 2 Ag A2 := fa2 : a 2 Ag ( ) X Xn 2 2 A := ai : a1; : : : ; an 2 A; n 2 N i=1 We will leave some fact below as an easy exercise for the reader 5 x0 Preliminaries De¯nition and Theorem 1. i. Given two porings (A; A+) and (B; B+), we write B+ extends A+ i® there exists a monomor- phism in the category of poring i :(A; A+) ! (B; B+) (i.e. i : A ! B is a ring monomor- phism and i(A+) ½ B+). ii. Any commutativeP unitary ring A that admits a partial ordering will also have the property that A2 is a partial ordering of it. For obvious reasons we may call this partial ordering the weakest partial ordering of A. We may refer this as the sums of squares of A as well. Speci¯cally, we have the following characterization: A admits a partial ordering i® Xn 2 2 8n 2 N; 8ai 2 A ai = 0 , ai = 0 8i = 1; : : : ; n i=0 iii. A poring (A; A+) induces a canonical partial order (resp. total order) · de¯ned by: x · y , y ¡ x 2 A+ By a reduced ring we mean that the only nilpotent element of the ring is 0. De¯nition and Theorem 2. i. (see [6] x1.3 p.38) Let (A; A+) be a poring and suppose that B is a ring that admits a partial ordering. If f : A ! B is a ring homomorphism, then there exists a partial ordering of B that contains f(A+) (thus making f a poring morphism) i® the set B+ ½ B de¯ned by Xn + 2 + B := f f(ai)bi : n 2 N; ai 2 A ; bi 2 B; i = 1; : : : ; ng i=1 is a partial ordering of B. If this occurs then B+ is called the weakest partial ordering of B that extends A+ (through f). Clearly here f(A+) ½ B+. ii. Let (A; A+) be a poring and S ½ A be a multiplicatively closed set. Then S¡1A admits a canonical partial ordering na o (S¡1A)+ := : 9t 2 S ¡ ast2 2 A+ s in such a way that the canonical ring homomorphism f : A ! S¡1A becomes poring morphism. In fact, it is easy to show that (S¡1A)+ is the weakest partial ordering of S¡1A that extends A+ through f. De¯nition. A ring A is called real i® the following identity holds for all n 2 N : 2 2 a1 + ¢ ¢ ¢ + an = 0 , a1; : : : ; an = 0 (8a1; : : : ; an 2 A) If A is a ¯eld then we call it a real ¯eld. Similarly we de¯ne real domains, real regular rings, ::: Remark. If A is a ring then being real implies the following ² A admits a partial ordering ² A is reduced Conversely, we note that if a ring A is a reduced poring then A is a real ring. De¯nition. Let F be a real ¯eld and suppose that for every real ¯eld K which is algebraic over F one has F = K, then we de¯ne F to be a real closed ¯eld. 6 x0 Preliminaries Theorem 3. Let F be a ¯eld, then the following are equivalent 1. F is a real closed ¯eld. 2. The set fx2 : x 2 F g is a total ordering of F and every polynomial of odd degree in F [T ] has a zero in F .

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