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Prime Spectrum of a Ring: a Survey THE PRIME SPECTRUM OF A RING: A SURVEY by James S. Fernandez A Thesis Submitted to the Faculty of the College of Science in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics Florida Atlantic University Boca Raton, Florida December 1991 THE PRIME SPECTRUM OF A RING: A SURVEY by James S. Fernandez This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Lee Klingler, Department of Mathematics, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the College of Science and was accepted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics. Supervisory Committee nt of Mathematics Date ii ACKNOWLEDGEMENTS I wish to express my sincere appreciation to my thesis advisor, Dr. Lee Klingler, for his patience and insight, and for lending assistance far beyond that which one normally assumes upon accepting the role of thesis advisor. I also wish to thank Dr. J . Brewer for his direction in getting this endeavor started and Dr. F. Richman for his insight into fine tuning the nuances during the final revision. I further wish to thank B. Broer, R. Pelava, and R. Ebel for their assistance and cooperation in getting this thesis printed. iii ABSTRACT AUTHOR James S. Fernandez TITLE The Prime Spectrum of a Ring: A Survey INSTITUTION Florida Atlantic University THESIS ADVISOR Dr. Lee Klingler DEGREE Master of Science in Mathematics YEAR 1991 This thesis has as its motivation the exploration, on an informal level, of a correspondence between Algebra and Topology. Specifically, it considers the prime spectrum of a ring, that is, the set of prime ideals, endowed with the Zariski topology. Questions posed by M. Atiyah and I. MacDonald in their book, "Introduction to Commutative Algebra", serve as a guideline through most of this work. The final section, however, follows R. Heitmann's paper, "Generating Non-Noetherian Modules Efficiently". This section examines the patch topology on the prime spectrum of a ring where the patch topology has as a closed subbasis the Zariski closed and Zariski quasi-compact open sets. It is proven that the prime spectrum of a ring with the patch topology is a compact Hausdorff space, and several relationships between the patch and Zariski topologies are established. The final section concludes with a technical theorem having a number of interesting corollaries, among which are a stable range theorem and a theorem of Kronecker, both generalized to the non-Noetherian setting. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS ........................................................... iii ABSTRACT ................................................................................. iv Section 1 Preliminaries: Some Properties of Prime Ideals ............................... 1 2 The Topology of the Prime Spectrum of a Ring ................. .............. 4 3 Irreducible Spaces .......................................................................... 8 4 Homomorphisms and Ring Spectra ................................................ 12 5 Localization and Ring Spectra ....................................................... 22 6 Integral Dependence ................................. .................................... 25 7 Noetherian Spaces ........................................................................ 32 8 The Patch Topology ..................................................................... 40 References . 53 v THE PRIME SPECTRUM OF A RING: A SURVEY §1 Preliminaries: Some Properties of Prime Ideals We begin our discussion by reviewing several definitions and propositions concerning the prime and radical ideals of a ring. The results proved in our discussion depend heavily on the fact that the rings involved are commutative and possess a multiplicative unity. Henceforth, all rings mentioned in the text are assumed to be commutative and possess a multiplicative unity. DEFINITION 1.1. The nilradical of a ring A is the subset consisting of all the nilpotent elements of A. For the sake of completeness we now recall, without proof, some standard properties of the prime and radical ideals of a ring A. PROPOSITION 1.2. Let A be a ring. 1) Any intersection of ideals of A, is an ideal of A; 2) The nilradical N of A is the intersection of all prime ideals contained in A. In particular, N is an ideal; 3) Let I be an ideal of A. The radical of I, denoted r(I), is defined to be the set {a E A I an E I for some n E IN}. The radical of I equals the intersection of all prime ideals in A which contain I. In particular, r(I) is an ideal of A. 1 4) Let P , ... , P n be prime ideals of A and let I be an ideal contained in 1 their union. Then I c Pi for some i. 5) Let I , ... , In be ideals of A and let P be a prime ideal containing their 1 intersection. Then Ii c P for some i. PROPOSITION 1.3. Let A be a ring, A 1 0. Let !fJ denote the set of all prime ideals of A. Then !fJ contains minimal elements with respect to inclusion. Proof Note that !fJ 1 ¢ since 0 is contained in some maximal, hence prime, ideal of A. Let C = { Pi I i E I} be a chain in !fJ where I is some indexing set. By invoking Zorn's lemma it suffices to show that i~ri E C. Since i~ri is the intersection of ideals of A, it is an ideal. We need only show i~ri is a prime ideal. Let a,b E A and suppose ab E i~ri" Without loss of generality, if b E Pi for all iEI then we're done, so suppose there exists kEI with b i Pk. Let j E I be such that P. c Pk. Then we have ab E P. and b; P . so that a E P. since P. J J J J J is a prime ideal. In particular, a E Pk. Hence for any Pi E I we have either Pi c Pk, in which case a E Pi by the above argument, or Pk c Pi in which case a E Pk n n c Pi. Thus a E iEli so that iEipi E C. Therefore, !fJ has a minimal element. C PROPOSITION 1.4. Let A be a ring, A 1 ¢, and let N be its nilradical. The following are equivalent: 1) A has exactly one prime ideal; 2) Every element of A is either a unit or a nilpotent element; 3) ~ is a field. 2 Proof We show {1) ~ {2) ~ {3) ~ {1). {1) ~ {2) : Suppose P is the sole prime ideal of A. Then P = N (prop. 1.3) and since a maximal ideal is also prime we know that P is the only maximal ideal of A. Hence, given x E A we have either x E P = N so that x is nilpotent; or x ;. P implying x is not contained in any maximal ideal of A so that x is a unit. {2) ~ {3): Assume {2) holds. Let I :/: (0) be an ideal of ~· Then there exists x;. N such that x+N E I. Since x ;. N, there exists y E A such that xy = 1. Thus (x+N)(y+N) = xy+N = 1+N E I. Hence I = {1) so that the only ideals of ~ are {0) and {1) . Thus ~ is a field. (3) ~ {1): Suppose ~ is a field. Then N is maximal ideal of A. Let P be any prime ideal of A. Then N c P. But N is maximal so N = P. Therefore, N is the only prime ideal of A. C DEFINITION 1.5. A ring having only one maximal ideal is called a local ring. PROPOSITION 1.6. Let A be a local ring with maximal ideal M. Then 1) Every element of A-M is a unit; 2) The only idempotents of A are 0 and 1. Proof 1) Since every non-unit of A is contained in some maximal ideal, (1) follows immediately. 2) Let a E A be an idempotent so that a(a-1) = 0 E M. Since M is a prime ideal, a E M or a-1 E M. If a E M then, since 1 ;. M, a-1 ;. M. Hence, a-1 is a unit so that a= 0. Similarly, a-1 E M ~ a-1 = 0. Therefore, a E {0,1 }. C 3 §2 The Tooology of the Prime Spectrum of a Ring Let A be a ring and let X denote the set of all prime ideals of A. Our goal is to endow X with a topology. To this end, for each subset E c A, let V(E) = { p E X I E ( p }. Suppose I is the ideal generated by E c A. Then since E c I c r(I), where r(I) denotes the radical of I, we clearly have V(r(I)) c V(I) c V(E). However, I is the smaUest ideal containing E so that P E V(E) ~ P E V(I). Hence, V(E) = V(I) . Also, r(I) equals the intersection of all prime ideals containing I so that V(r(I)) = V(I). Therefore, for any subset E c A we have V(E) = V(I) = V(r(I)). Consider the cases when E = (0) or E = (1). Since 0 E P, for all P EX, we certainly have V(O) = X. Now, if P E X, then 1 ;. P so that V(1) = ¢. Let (Ei)iE/ be any family of subsets of A (I some indexing set). Let P E V(i~/Ei) . Then i~.zEi c P so that Ei c P, for each i E I. Thus P E V(Ei), each i E I and hence P E i~/V(Ei). Therefore, V(i~.zEi) c i~/V(Ei). Inspection shows this argument can be reversed to give i~/V(Ei) c V(i~/Ei).
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