CALIFORNIA STATE UNIVERSITY, NORTHRIDGE The Zariski Topology on the Prime Spectrum of a Commutative Ring A thesis submitted in partial fulllment of the requirements for the degree of Master of Science in Mathematics by Choomee Kim December 2018 The thesis of Choomee Kim is approved: Jason Lo, Ph.D. Date Katherine Stevenson, Ph.D. Date Jerry Rosen, Ph.D., Chair Date California State University, Northridge ii Dedication I dedicate this to my beloved family. iii Acknowledgments With deep gratitude and appreciation, I would like to rst acknowledge and thank my thesis advisor, Professor Jerry Rosen, for all of the support and guidance he provided during the past two years of my graduate study. I would like to extend my sincere appreciation to Professor Jason Lo and Professor Katherine Stevenson for serving on my thesis committee and taking time to read and advise. I would also like to thank the CSUN faculty and cohort, especially Professor Mary Rosen, Yen Doung, and all of my oce colleagues, for their genuine friendship, encouragement, and emotional support throughout the course of my graduate study. All this has been truly enriching experience, and I would like to give back to our students and colleagues as to become a professor who cares for others around us. Last, but not least, I would like to thank my husband and my daughter, Darlene, for their emotional and moral support, love, and encouragement, and without which this would not have been possible. iv Table of Contents Signature Page.................................... ii Dedication...................................... iii Acknowledgements.................................. iv Table of Contents.................................. v Abstract........................................ vii 1 Preliminaries................................... 1 1.1 Commutative Rings..........................1 1.1.1 Basic Notions........................1 1.1.2 Noetherian Rings......................6 1.1.3 Localization.........................8 1.2 Topology................................ 10 2 An Introduction to the Zariski Topology on the Prime Spectrum of R . 14 2.1 Introduction.............................. 14 2.2 Mappings................................ 20 2.3 Noetherian Topological Spaces.................... 23 3 Connectedness and Dimension in Spec(R) ................... 25 3.1 Connectedness in Spec(R) ....................... 25 3.2 Krull Dimension............................ 28 3.3 Irreducibility.............................. 34 4 Worked Examples................................ 36 4.1 ................................ 36 Spec(Zn) v 4.2 Spec(Z[x]) ................................ 37 4.3 Spec of C(X) .............................. 40 4.4 Spec of a Boolean Ring........................ 45 Bibliography..................................... 50 vi Abstract The Zariski Topology on the Prime Spectrum of a Commutative Ring by Choomee Kim Master of Science in Mathematics The Zariski Topology is an interesting topic in algebraic geometry that combines commutative algebra and topology to deal with questions that are algebraic and ge- ometrical in nature. It was introduced by Oscar Zariski in the mid twentieth century and has had great success helping to answer several long standing, famous questions, such as the solution(by Andrew Wiles in the 1990s) to Fermat's famous last theorem. Initially, algebraic geometry dealt with polynomial rings in nitely many variables over a eld. A great deal of research went to understanding the qualitative and quan- titative nature of solutions to polynomial equations in several variables. As is the case in studying rings, the understanding the ideal structure of F [x1; x2; ··· ; xn] where F is a eld of characteristic zero was of paramount importance. Initially, the maximal ideal structure of such rings played a vital role in algebraic geometry, and Hilbert's Nullstellensats completely characterized the maximal ideal of F [x1; x2; ··· ; xn] when F has characteristic zero. In particular, the maximal ideals of this ring are of the form (x1 − a1; ··· ; xn − an) where ai 2 F . The signicance of this result is that it provides a natural one to one correspondence between the maximal ideals of F [x1; x1; ··· ; xn] and the points in ane n-space F n. However, one drawback to dening topologies in vii terms of spaces of maximal ideals is that under a homomorphism, from a commutative ring to itself, the inverse image of a maximal ideal may not be maximal. The problem with this is that ring homomorphisms, when extended to the maximal spectrum of the ring, may fail to be continuous. Hence, the maximal spectrum, while important, may not be the best topology to consider. Eventually, Zariski and others realized that the ideal topology is one that is dened using the prime ideals of a commutative ring. This topology is called the prime spectrum of the commutative ring, and it has the maximal spectrum as a topological subspace. The primary purpose of this thesis is to investigate the prime spectrum of commutative ring. We will prove some general results about this spectrum and then specialize to specic examples. For instance, we will investigate the prime spectrum of Z[x], Z the ring of integers, and C(X), the ring of continuous functions on a compact Hausdor space X. One interesting feature of the Zariski topology on a commutative ring is that it often produces surprising prop- erties. For example, it is always compact, but rarely Hausdor. We will, however, give necessary and sucient conditions on a ring, which result in this topology being Hausdor. We will also discuss connectedness and see that properties on the ring will lead to the topology being connected. Hence, the aim of our thesis is to simply investigate the connection between a prime ring R and the topological properties of Spec(R). viii Chapter 1 Preliminaries 1.1 Commutative Rings 1.1.1 Basic Notions In this thesis, all rings are assumed to be commutative. Denition 1.1. A subset I of a ring R is called an ideal if it is closed under addition, contains 0 and satises the ideal property. That is for any a 2 I and r 2 R, ar 2 I. Denition 1.2. An ideal P 6= R is called a prime ideal if ab 2 P implies a 2 P or b 2 P . An ideal M 6= R is called a maximal ideal if M ⊆ I E R, then either M = I or I = R. Proposition 1.1. 1. An ideal P 6= R is a prime ideal if and only if R=P is an integral domain. 2. An ideal M 6= R is a maximal ideal if and only if R=M is a eld. Proof. 1. Assume P 6= R is a prime ideal and let (a + P )(b + P ) = ab + P = 0 + P . Then ab 2 P and so a 2 P or b 2 P implies a + P = 0 + P or b + P = 0 + P . Thus R=P is an integral domain. Now assume R=P is an integral domain and let (a + P )(b + P ) = ab + P = 0 + P . Then ab 2 P and either a + P = 0 + P or b + P = 0 + P which implies a 2 P or b 2 P . Thus P is a prime ideal. 2. First we assume M is a maximal ideal of R. If r + M 6= 0 + M, then r2 = M and by maximality we have M + (r) = R. Then 1 = m + ar for some m 2 M and a 2 R which gives 1 + M = m + ar + M = ar + M = (a + M)(r + M) 1 proving r + M is invertible. Now we assume R=M is eld and let M ( I E R. If a 2 I − M, then a + M 6= 0 + M and there exists b + M 2 R=M such that 1 + M = ab + M, since R=M is eld. Hence 1 − ab 2 M ( I and since ab 2 I, we obtain 1 2 I. This implies I = R so M is maximal ideal. Denition 1.3. An integral domain R is called a Principal Ideal Domain (PID) if every ideal is principal of the form (a) = aR Proposition 1.2. In a PID, every nonzero prime ideal is maximal. Proof. Let P 6= (0) be prime ideal and P ( I E R. Since it is PID, we let P = (a) ( I = (b) E R. Then a 2 (a) ( (b) gives a = bc 2 (a), so we have b 2 (a) or c 2 (a) but b2 = (a) by our assumption. Thus c = ar = bcr and 1 = br, showing (b) = R, so I = R. Theorem 1.3. (First Isomorphism Theorem) If f : R ! S is a ring epimorphism, then R=Ker(f) ∼= S. Proof. Dene f~ : R=ker(f) ! S by f~(r + Ker(f)) = f(r). It is easy to check f~ is well dened ring homomorphism. If r + Ker(f) 2 Ker(f~) if and only if 0 + Ker(f) = f~(r + Ker(f)) = f(r) if and only if r 2 Ker(f) if and only if r + Ker(f) = 0. Hence f~ is also one-to-one proving it is isomorphism. Theorem 1.4. (Correspondence Theorem) For given onto homomorphism f : R ! S, f induces a bijection between the ideal of R containing Ker(f) and all the ideal of S. −1 −1 Proof. Let J/S and we have f0sg ⊆ J. Then Ker(f) = f (f0sg) ⊂ f (J) /R and f(f −1(J)) = J since f is onto proving it is onto. To prove one-to-one, we suppose A; B are ideals of R containing Ker(f) and f(A) = f(B). For any a 2 A, f(a) 2 f(B), 2 so there exists b 2 B such that f(a) = f(b). Then f(a) − f(b) = f(a − b) = 0 by homomorphism, so a − b 2 Ker(f) ⊆ B showing a 2 B, proving A ⊆ B. Analogously B ⊆ A and so we have a one-to-one correspondence. It is important to investigate what happens to prime and maximal ideals under this correspondence theorem.