Polynomial Extensions of Commutative Rings By David A. Lieberman A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M a s t e r s in Sc ie n c e (M a t h e m a t ic s ) at the CALIFORNIA STATE UNIVERSITY - CHANNEL ISLANDS 2018 (C 2018 David Andrew Lieberman ALL RIGHTS RESERVED APPROVED FOR THE MATHEMATICS PROGRAM Dr. Jesse Elliott, Advisor Datr May 31, 2018 Dr. Roger Roybal Date May 2018 28, Dr. Brian Sittinger Date May 28 2018 A P P ROVED FOR THE UNIVERSITY Dr. Joseph Shapiro, AVP Extended University Date May 31, 2018 Acknowledgements This work could not have been done without the advisement and aid of my advisor, Dr. Jesse Elliott, from California State University Channel Islands. Thank you to Dr. Brian Sittinger and Dr. Roger Roybal for taking the time to be members of my thesis committee. Camarillo, California June 1, 2018 iv " “You miss 100% of the shots you don’t take. -Wayne Gretzky” " -Michael Scott -David Lieberman 1 Abstract This paper aims to generalize results on single variable polynomial rings over commutative rings with zerodivisors to the case of polynomial rings in arbitrarily many variables. Given a commutative R,ring we give necessary and sufficient conditions for the ring of polynomials with coefficients in R in arbitrarily many variables to be a PVMR and Krull ring. In answering these questions, we make use of the t and v operations on ideals as a means of characterizing these rings. We also give conjectures on necessary and sufficient conditions for an arbitrary polynomial ring to be a Dedekind ring, a UFR, and integrally closed. 2 Table of Contents 1 Introduction 4 2 Background 7 2.1 Integral D om a in s......................................................................... 7 2.2 Some Useful Results On D om ain s............................................. 14 2.3 Commutative Rings with Zerodivisors...................................... 15 2.4 Semistar Operations ................................................................... 19 2.5 Some Results Due to H uckaba................................................... 23 2.6 Some Results Due to E llio t t ...................................................... 26 2.7 Polynomial Ring Extensions in a Single Variable................... 27 3 Arbitrary Polynomial Extensions 31 3.1 When is R[Xx] a PVMR? ......................................................... 32 3.2 When is R[Xx] a Krull Ring? ................................................... 36 3.3 Future W ork................................................................................... 38 3 1 Introduction In this paper, we investigate properties of polynomial extensions of commu­ tative rings with unity. We begin by giving a brief overview of the basics of ideal theory with respect to integral domains. This will motivate the work done in subsequent chapters on arbitrary commutative rings with zerodivi- sors. The background work we discuss in Chapter 2 will allow us to arrive at our main results in chapter 3. We begin Chapter 2 by giving a brief overview of multiplicative ideal theory as it relates to integral domains. Specifically, we focus on the use star-operations (in particular thed, v, t, and w operations) to give new and alternative definitions to classes of integral domains. As it turns out, star- operations work as a powerful tool in providing definitions of the different types of integral domains in a way that connects these classes together. This allows us to give definitions for Prufer domains, Krull domains, Prufer v-Multiplication domains, UFD’s, and Dedekind domains. 4 CHAPTER 1. INTRODUCTION We continue Chapter 2 by generalizing much of what we show concerning integral domains to the realm of commutative rings. After finding analogous definitions for quotient fields and fractional ideals, we are able to generalize each of the previously discussed classes of integral domains to their com­ mutative rings (with zerodivisors) counterparts. This is done quite readily by generalizing the notion of star-operations on the fractional ideals of a domain to the notion of semistar-operations on the Kaplansky fractional ideals of a commutative ring. Semistar-operations lend their hand to defin­ ing classes of commutative rings in much the same way as star-operations do for domains. This tool will allow us to give analogous definitions for Prufer rings, Krull rings, Prufer v-Multiplication rings, UFr’s, Dedekind rings, and others. We conclude Chapter 2 with a review of known results specifically con­ cerning the case of polynomial rings in a single variable over a commutative ring. For a commutative ringR, necessary and sufficient conditions have been studied and found for the ring R [X ] to be a Krull ring, a Prufer v-Multiplication ring, or a unique factorization ring (see [3], [12], and [4] respectively). We recount these results at the end of Chapter 2. In Chapter 3, we tackle the main goal of this paper: given an arbitrary 5 CHAPTER 1. INTRODUCTION set of determinates X\ = ( X aI aga and a commutative ringR, what are the necessary and sufficient conditions forR [X a] to be a PVMR, a Krull ring, a Dedekind ring, a UFR, or integrally closed, and, in particular, we succeed in proving the following two theorems: Theorem 1.0.1. The ring R [X a] is a PVMR if and only if R is a PVMR and T (R) is von Neumann regular. Theorem 1.0.2. If R is a commutative ring, then R[X a] is a Krull ring if and only if R is a finite product of Krull domains. We also present some progress made on finding necessary and sufficient conditions for R [X a] to be a Dedekind ring, and we present some conjectures pertaining to the case of UFRs and integral closedness. 6 2 Background 2.1 Integral Domains As a motivation for what is to follow in our treatment of commutative rings with zerodivisors, we first examine some properties of integral domains. A useful tool for the study of integral domains is the concept of star operations. Star operations, originally introduced by Krull in [6] as ’-operations, are self-maps on the set of fractional ideals of a domainD, which we denote as F (D). For the remainder of this chapter, D will be an integral domain with quotient field K . We also will, on occasion, referintegral domains simply as domains. Definition 2.1.1. A fractional ideal of D is a D-submodule I of K such that, for some nonzero a G D, a l C D. We say that I is invertible if there is some J G F (D) such that IJ = D. 7 C H A P T E R 2. B A C K G R O U N D A useful notation for our work moving forward will be that of the ideal quotient (also called the colon ideal). Given a fractional ideal I of a domain D, we are able to define a generalized notion of the inverse of an ideal. Definition 2.1.2. Let I be a fractional ideal of an integral domain D, with quotient field K . We define (D :K I ) = {x G K : x I C D }, which we denote as I-1. Proposition 2.1.3. If I G F (D) is invertible, then I I-1 = D. Proof. Let I G F (D ) be invertible, with IJ = D for some J G F(D). Then for any arbitrarily fixedj G J , we have ij G D for all i G I . Thus j G I -1 , and furthermore J C I -1 . Therefore IJ = D C I I -1 . Observe that, by definition, x I C D for all x G I -1 . Thus I I -1 C D. Therefore we conclude that I I -1 = D. ■ We continue by defining star operations on the fractional ideals of on integral domain D. These will prove to be a valuable tool for our work mov­ ing forward. Namely, they will serve to classify different types of integral domains through properties of their fractional ideals. 8 2.1. INTEGRAL DOMAINS Definition 2.1.4. A star operation on an integral domain D is a unary operation * : F (D) ^ F (D), with I ^ ) := I * such that for all fractional ideals I, J E F(D), i. ) I C J implies I * C J * (order preserving), ii. ) I C I * (expansive), iii. ) (I*)* = I * (idempotent), iv. ) I* J* C (IJ)* (sub-multiplicative), v.) D* = D (unital). Definition 2.1.5. We .say a fractional ideal I E F(D) is ^-closed if I satisfies the equation I = I *, and we say I is ^-invertible if there is some J E F(D) such that (IJ)* = D. The specific example of a star operation that we make note of is the trivial star operation, which is called thed operation. That is, for a frac­ tional ideal of an integral domain D, we define the mapd : F(D) ^ F(D) by d(I) = I, which we denote by d(I) = I d. Within the scope of this pa­ per, the two star operations of greatest use will be the t-operation and the v-operation, which we define along with the w-operation. 9 CHAPTER 2. BACKGROUND For the following definitions, let I be a fractional ideal of an integral domain D Definition 2.1.6. The v-operation, or v-closure star operation is the map v : F (D) ^ F(D), denoted by v (I) = I'v, and defined by I V = ( / - i)- i. Definition 2.1.7. The t-operation, or t-closure star operation is the map t : F (D) ^ F(D), denoted by t(I) = I*, and defined by I * ^ ^ {J v : J G F (D) is finitely generated and J C I } . Definition 2.1.8. The w-operation, or w -closure star operation is the map w : F(D) ^ F (D), denoted by w (I) = I w, and defined by Iw = U {(I J) : J G F(D) is finitely generated and Jv = D }.