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 May2018 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 }.

Using the above star operations, we can define several important classes of integral domains. What follows is a sampling of the ways in which star operations yield useful equivalent characterizations of these classes of do­ mains, as given in [13].

10 2.1. INTEGRAL DOMAINS

Definition 2.1.9. A domain D is a P ru fer v-M ultiplication dom ain

(abbreviated P V M D ) if all nonzero finitely generated fractional ideals are t-invertible.

Definition 2.1.10. A domain D is a P ru fer dom ain if D satisfies either of the following equivalent characterizations:

i. ) all nonzero finitely generated fractional ideals are invertible

ii. ) D is a PVMD and A* = for all fractional ideals A of D.

Proof of Equivalence. Suppose that all nonzero finitely generated fractional ideals of D are invertible. If I GF (D) is a nonzero finitely generated fractional ideal, thenI I -1 = D. Since t is unital and order preserving, we see that (I I -1)* = D* = D. Thus all nonzero finitely generated ideals are t-invertible. Therefore D is a PVMD.

Let A G F (D ), where D is again a domain in which all nonzero finitely generated fractional ideals are invertible. LetJ be a nonzero finitely gen­ erated (fractional) ideal of D which is contained in A. By assumption, J is

11 CHAPTER 2. BACKGROUND invertible. Thus J is v-closed [8, Proposition 34.12]. Therefore we have

A* = ^ { Jv : J G F (D) is finitely generated and J CA}

^ ^ {J : J G F (D) is finitely generated and J C A }

Since each of the above J is contained in A, we know their union is as well.

Thus we know A* C A. Because the t-operation is order preserving, we also have A C A*. Therefore A* = A = Ad for all A GF(D).

Conversely, suppose D is a PVMD and I * = I d for all fractional ideals of

D. Then if I is a nonzero finitely generated fractional ideal of D, we know that (IJ)* = D for some J G F (D ). By assumption, (IJ )* = (IJ )d = I J .

Thus I is invertible, and therefore all nonzero finitely generated fractional ideals of D are invertible. ■

Definition 2.1.11. A domain is M ori if it satisfies the ascending chain condition (ACC) on v-closed (fractional) ideals.

Definition 2.1.12. [5,13] A domain D is a Krull domain if D satisfies either of the following equivalent characterizations:

i.) all nonzero (fractional) ideals are t-invertible

12 2.1. INTEGRAL DOMAINS

ii.) D is a Mori PVMD.

Definition 2.1.13. A domain D is a Dedekind dom ain if D satisfies any one of the following equivalent characterizations:

i. ) every nonzero proper ideal factors into a product of prime ideals

ii. ) all nonzero (fractional) ideals are invertible.

Definition 2.1.14. An integral domain is a unique fa ctoriza tion do­ main, or UFD, if every nonzero element can be written as a product of a unit and prime elements of the domain. Equivalently, a UFD is a domain in which all t-closed ideals are principal.

13 CHAPTER 2. BACKGROUND

2.2 Some Useful Results On Domains

We close this chapter with two useful results which will be of use to us in subsequent chapters. The first results comes from Bourbaki.

P roposition 2.2.1. [1, Proposition VII.1.9] If D is a KruU domain, then

D[X] is a KruU domain.

Corollary 2.2.2. If D is a KruU domain, then D [X i,...,X n] is a KruU domain (where n G N is fixed).

14 2.3. COMMUTATIVE RINGS WITH ZERODIVISORS

2.3 Com m utative Rings with Zerodivisors

In this section, we generalize notions and concepts related to integral do­ mains from the previous section to commutative rings with zerodivisors.

We begin by generalizing the objects and characterizations pertaining to domains to the case of commutative rings with zerodivisors. For the rest of this and subsequent chapters we assume that all rings are commutative and have a multiplicative identity.

Definition 2.3.1. A total quotient ring is a ring in which every element that is not a zerodivisor is a unit. Given a ring R, let U be the multiplicative set of all non-zerodivisors of R. We define T(R) := U-1 R, which we call the total quotient ring of R. That is

T (R) = |“/ b = [(a, b)] :a,b G R, and b G U} , is a set of equivalence classes defined by

(a, b) ~ (c, d) ad = bc.

Definition 2.3.2. For a ring R with total quotient ring T(R), the R- submodules of T(R) form the set of Kaplansky fractional ideals of R, which we denote by K(R).

15 CHAPTER 2. BACKGROUND

Definition 2.3.3. For a subring R of a ring T , and R-submodules I and

J of T , define (I :T J) = {x G T : xJ C I }. When the context is clear, we may interchangeably use (I :T J) = (I : J) (for example, when T =T(R)).

This notation precisely mirrors the colon notation for domains, and it will serve as a useful way to develop a generalized concept of inverses in commutative rings with zerodivisors.

Definition 2.3.4. Given a ring R and some I G K(R), we define the following notation:

I -1 = (R :t(r) I ) = {x G T(R) : xI C R}.

Definition 2.3.5. Let R be a ring. We say an element of R is regular if it is non-zerodivisor, and an element I G K is regular if I contains a regular element.

Definition 2.3.6. Let R be a ring with total quotient ring T(R). We say that I G K(R) is fractional or a fractional ideal if -1I is regular, that is, if there is a regular element of r G R such that rI C R

Remark 2.3.7. Note that a fractional ideal need not be an ideal in the traditional sense, and a Kaplansky fractional ideal need not be fractional in

16 2.3. COMMUTATIVE RINGS WITH ZERODIVISORS

the above sense. These notions serve to generalize the concepts of ideals

and fractional ideals, respectively.

Definition 2.3.8. [13, Proposition 2.1.23] For a ring R and a Kaplansky fractional ideal I E K(R), we .say I is invertible if there exists some J E

T (R) such that IJ = R.

Proposition 2.3.9. [13, Proposition 2.1.23] For a ring R, if I E K(R) is

invertible, then I I -1 = R. Furthermore I is finitely generated and regular.

Proof. Let I E K(R) be invertible. Then there is some J E K(R) such that

IJ = R. First observe that, by definition, for allx I C R for all x E I .

Thus I I -1 C R. It suffices to show that R C I I -1 . Since IJ = R, we know

that if x E J , then xy E R for all y E I . Thus for any such x, we have

x I C R. Therefore J C I . Let r be an arbitrary element of R. Then as

IJ = R, there are some elements xi,... ,xn E J and yi,... ,yn such that

r ^ ^n=1 x^y^. Since J C I -1 , each x^ is also an element of I -1 . Thus the

sum is also an element of I I -1 , and thus R C I I -1 . Therefore I I -1 = R.

We now show that I is finitely generated. Since I I -1 = R, we know

that 1 E I I -1 , and thus for some finite sequence of elementsm1,..., mk E I

17 CHAPTER 2. BACKGROUND and ui,... ,Uk G I 1, k "Y^miUi = 1. i=l

Then for any x G I , we have

k x = x^ ^ miui i=l k ^ xmiui i=l k ^ ^ ai'mi, i=l with ai = xui G T(R ). Therefore I is finitely generated byml,..., mk.

We now show that I is regular. Letml,..., mk G I and ul,..., uk G I -1 be elements such tha^ ^ k=l miui = 1. Since each element of I -1 is an element of T (R), there are some elements al ... ,ak G R and some regular elements bl ... ,bk G R such that ui = ai/bi for all i G |1,...,fc}. Let b = bl ■ ... ■ bn, and note that b is regular. Then we have

k k b = b ■ 1 = b miui ^ mi(bni). i=l i=l

Each term mi(bni) of this sum is an element of I , as it is a product of an element of I and an element ofR . Thus the sum itself is an element of I.

Thus I contains the regular element b, and therefore I is regular. ■

18 2.4 SEMISTAR OPERATIONS

2.4 Semistar Operations

Just as star operations can be used to gain a handle on the ideals of integral domains, the generalized notion of semistar operations can be used to bet­ ter understand the ideal structure of commutative rings with zerodivisors.

What follows in this section is found in [13, Chapter 2].

D efinition 2.4.1. [13, Definition 2.3.1] Let R be a ring and I G K(R). A sem istar operation on R is a self-map ^ : I ^ I* such that -k is order­ preserving, expansive, idempotent, sub-multiplicative, and unital. That is, k satisfies the following for any I, J G K(R):

1. If I C J , then I* C J*,

2. I C I*,

3. (I*)* = I*,

4. I*J* c (IJ)*,

5. R* = R.

Definition 2.4.2. For a ring R and some I G K(R), we say I is k- invertible if there exists some J G K(R) such that (IJ)* = R. We say that I k-closed if I* = I .

19 CHAPTER 2. BACKGROUND

Much like in the case of integral domains, two extremely useful semistar operations for our work will be the v-operation and t-operation, which are defined in similar fashion. In fact these operations require no altering to apply to rings with zerodivisors, other than its application to K(R) rather than just F(R).

Definition 2.4.3. The v-operation is a semistar operation is defined by the map v : K(R) ^ K(R), with v (I) = (I-1)-1 . We denote v (I) = Tv

Definition 2.4.4. The t-closure is a semistar operation is defined by the map t : K(R) ^ K(R), with

t(i { • j : J G K(R) is finitely generated and J C I } .

We denote t(I) = I*.

Definition 2.4.5. The d-closure operation is the trivial semistar opera­ tion, with I d = I for all I G K(R).

The semistar operations we have defined allow us to generalize the clas­ sifications of integral domains to the realm of commutative rings. We first generalize the notions of Dedekind and Prufer domains.

20 24- SEMISTAR OPERATIONS

D efinition 2.4.6. [13, Definition 2.4.4] Let R be a ring with a sewistar operation -k.

1. R is -k-Dedekind if all regular (fractional) ideals of R are k-invertible.

2. R is k -P ru fer if all finitely generated regular (fractional) ideals of R

are k-invertible.

Using this notion along with the t,v,w, and d operations, we can define generalizations of the types of integral domains to commutative rings.

D efinition 2.4.7. [13, pg. 35]

1. A Dedekind ring is a d-Dedekind ring. That is, a ring in which all

regular (fractional) ideals of R are d-invertible, or rather invertible.

2. A K rull ring is a t-Dedekind ring. That is, a ring in which all regular

(fractional) ideals of R are t-invertible

3. A P ru fer ring is a d-Prufer ring. That is, a ring in which all finitely

generated regular (fractional) ideals are d-invertible, or rather invert­

ible.

21 CHAPTER 2. BACKGROUND

4. A Prufer v-multiplication ring, or PVMR, is a d-Prufer ring.

That is, a ring in which all finitely generated regular (fractional) ideals

are d-invertible.

D efinition 2.4.8. [13, pg. 90] A ring is a unique factorization ring, or

UFR, if every non-unit is a product of prime elements. Equivalently it is a ring which is isomorphic to a finite direct product of UFDs and principal ideal rings.

22 2.5. SOME RESULTS DUE TO HUCKABA

2.5 Some Results Due to Huckaba

In order to prove some of our results in subsequent sections we recall some results given by Huckaba in [9]. These results concern the following prop­ erties of rings.

Definition 2.5.1. A ring R is von Neumann regular if for any element r E R, there exists an x E R such that r2 x = r.

Definition 2.5.2. A commutative ring R is said to satisfy P rop ery A if any finitely generated ideal of R comprising only zerodivisors has a non zero annihilator. Equivalently R satisfies property A if all finitely generated dense ideals are regular. An ideal A of R is dense if, for all r E R, r l= 0 if and only if r = 0.

Recall that a ring is said to be reduced if it has no nonzero nilpotent elements.

Proposition 2.5.3. Let R be a commutative ring with unity. If R is von

Neumann regular, then R is reduced.

Proof. Assume to a contradiction that for some non-zero r E R, there exists a minimaln E N>i such that rn = 0. Since R is von Neumann regular,

23 CHAPTER 2. BACKGROUND there is some x E R such that xr2 = r. If n = 2 , then r = r2x = 0, which is a contradiction. If n > 2, then rn-1 = (rn-2)(r2x) = rnx = 0.

This contradicts the minimality of n. Thus there are no nonzero nilpotent elements of R. ■

Corollary 2.5.4. If R is a ring and T(R) is von Neumann regular, then

R is reduced.

We now give two important results from [9] which we then use to prove

Corollary 2.5.7.

Theorem 2.5.5. [9, Theorem 4.5] Let R be a reduced ring. Then T(R) is von Neuman regular if and only if Min R is compact with respect to the

Zariski topology, and R satisfies Property A.

Corollary 2.5.6. [9, Corollary 4.6] Let R be a reduced ring. Then Min R is compact if and only ifT(R[X ]) is a von Neumann regular ring.

Corollary 2.5.7. Let R be a reduced ring. Then T(R) is von Neumann regular if and only ifT(R[X ]) is von Neumann regular.

Proof. Let R be a reduced commutative ring, and suppose that(R[X T ]) is von Neumann regular. As T(R ) C T (R [X ]), we know that for any r a/b= E

24 2.5. SOME RESULTS DUE TO HUCKABA

T (R), there is some f / g G T (R[X]) such that g (r)2 = r. However since the right hand side of this equality is a degree zero polynomial, we deduce that f / g must be an element of T(R ). Therefore T(R ) is von Neumann regular.

Let R be a reduced ring, and suppose that T(R ) is von Neumann regular.

Then by Theorem 2.5.5, Min R is compact. Therefore by Corollary 2.5.6

T (R [X ]) is von Neumann regular. ■

25 CHAPTER 2. BACKGROUND

2.6 Some Results Due to Elliott

What follows are some useful results which given by Elliott in [13]. These propositions are concerned with lifting ideals from a ringR and viewing them as ideals in the context of some extension of the ring R. Of particular importance for our work are the results on t-linked extensions:

Definition 2.6.1. Let R be a ring and S a ring extension of R. We say S is a t-linked exten sion of R if I* = R implies (IS)* = S for all ideals I of R.

Proposition 2.6.2. [13, Proposition 2.3.1] Let R be ring, and let S be an

R-torsion-free R-algebra. If S is flat as an R-module then S is a t-linked extension of R.

Since any free module is both flat and torsion-free, this gives the follow­ ing corollary.

Corollary 2.6.3. Let R and S be rings such that R C S . If S is free as an

R-module, then S is a t-linked extension of R.

26 2.7. POLYNOMIAL RING EXTENSIONS IN A SINGLE VARIABLE

2.7 Polynom ial Ring Extensions in a Single

V a r i a b l e

We conclude this chapter by presenting known results on polynomial rings in a single variable. These results will be expanded to the case of arbitrarily many variables in Chapter 3. The first theorem we present was first shown by Lucas in 2005.

Theorem 2.7.1. [12, Theorem 7.8] For a ring R, the following are equiv­ alent:

(1) R [X ] is a PVMR

(2) R is a PVMR and T(R) is von Neumann regular.

Theorem 2.7.2. [3, Theorem 5.7] If R is a commutative ring, then R [X ] is a Krull ring if and only if R is a finitedirect product of Krull domains.

Proof. The proof of the reverse implication (that R is a finite direct product of Krull domains implies R[X] is a Krull ring) can be gleaned from the proof of Theorem 3.2.1 in Section 3.2. Thus we only give proof of the forward implication at this time. Additionally part of this proof as given in [3] requires a knowledge of the primary decomposition of ideals that is

27 CHAPTER 2. BACKGROUND currently beyond the scope of this paper. Thus we give a sketch of the proof where appropriate.

Suppose that R[X] is a Krull ring. Observe thatR must then be reduced as follows: assume to the contrary that there is some nilpotent r G R with rn = 0 for some fixed natural numbern. Then the element G T (R [X ]) is integral over R, as it is a root of the polynomial G (R[X])[Y]. This is absurd, as all Krull rings are integrally closed. Thus R is reduced.

Through primary decomposition of the principal regular ideal (X ), An­ derson, Anderson, and Markanda show in [3] that R has finitely many min­ imal prime ideals, sayPi,..., Pn. Furthermore it is shown that (0) is equal to the intersection of these prime ideals. Thus R C R /P l x ■ ■ ■ x R /P n, where each R/Pi is an integral domain. As this product is a finitely gener­ ated R-module and R is integrally closed, R is in fact equal to the product of quotient rings. Therefore as R is a Krull ring, we also know that R /P i is a Krull domain fori = 1,..., n. ■

In [3] the following is given as a corollary to the above theorem, and is also proved separately in [4]. Note that while [4] was found by the authors to have errors, in the subsequently published corrigendum ([2]) the stated errors do not effect the proof of the following theorem.

28 2.7. POLYNOMIAL RING EXTENSIONS IN A SINGLE VARIABLE

Theorem 2.7.3. [3, Corollary 5.8] If R is a commutative ring, then R [X] is a UFR if and only if R is a finite direct product of UFDs.

In [10], Lucas proved the following necessary and sufficient conditions for determining when R [X ] is integrally closed.

Theorem 2.7.4. [10, Corollary 5] Let R be a commutative ring. Then

R [X ] is integrally closed (in T(R [X ])) if and only if R is integrally closed in T(R[X]).

Lucas presents this result as a corollary to the following proposition.

Proposition 2.7.5. [10, Theorem 4] Let R be an integrally closed (inT(R)) and reduced ring. Then R [X] is not integrally closed if and only if there exists an element f/g E T(R[X]) \ R [X ] that is integral over R.

We proceed with the showing that Theorem 2.7.4 follows from Proposi­ tion 2.7.5.

Proof. If R is both integrally closed in T(R ) and reduced, then the theorem is the contrapositive of the equivalence of (1) and (2) from Proposition 2.7.5.

Clearly the negation of (1) is “ R [X ] is integrally closed.” The negation of

(2) is “there is no element of T (R [X ]) \ R [X ] which integral over R.” In

29 CHAPTER 2. BACKGROUND such a case it is also true that “there is no element of(R[X T ]) \ R which integral over R.” This is the definition of R is integrally closed in T(R[X]).

For the case when R is either not reduced or not integrally closed, we show that R [X ] is not integrally closed and R is not integrally closed in

T (R [X ]), thus maintaining the validity of the theorem.

If R is not integrally closed in T(R ), then there is some element a G

T(R ) \ R which is integral over R. Since T(R ) C T (R [X ]), it is also the case thata G T (R [X ]) \ R. In other words, a is integral over R, and it is an element of T (R [X ]) which is not an element of R. Therefore R is not integrally closed in T (R [X ]). Furthermore observe that a G R [X ] and a is integral over R [X ]. Thus R [X ] is not integrally closed.

For the case when R is not reduced, it suffices to show that R is not integrally closed in T(R ). If R is not reduced, then there exists some nilpo- tent element r G R such that rn = 0 for some n G N. Consider the element r /X G T (R [X ]). Observe that r /X is a root of xn, which is a monic poly­ nomial over R. Thus r /X is integral over R. However since r /X is not a polynomial, it is not an element of R [X ]. Therefore the integral closure of

R in T(R) is not equal to R. In other words R is not integrally closed. ■

30 3

Arbitrary Polynomial Extensions

In this chapter, (X aI aga denotes an arbitrary set of indeterminates indexed by some indexing set A. Note that we allow A to be uncountable. As usual, any ringR will be assumed to be commutative ring with unity and with total quotient ring T(R ). We consider the necessary and sufficient conditions required for the ring of polynomials with variables in|X a} and coefficients in R to have some of the previously mentioned properties of commutative rings. We use the notationR[Xx] = R[{Xx}] to denote such a polynomial ring.

31 CHAPTER 3. ARBITRARY POLYNOMIAL EXTENSIONS

3.1 W hen is R[X a] a P V M R ?

We begin by giving a corollary to Theorem 2.7.1. The use of this corollary, and the lemma that follows, will be crucial in showing when an arbitrary polynomial ring is a PVMR.

Corollary 3.1.1. Let R be a ring and let X \ ,X 2, ... ,X n be indeterminates for some fixed n G N. Then R[Xi,... ,Xn] is a PVMR if and only if R is a PVMR and T (R) is von Neumann regular.

Proof. Suppose R is a PVMR and T(R ) is von Neumann regular, and note that by Corollary 2.5.4, R is reduced. Then by Theorem 2.7.1, R [X 1] is a

PVMR. Furthermore by Corollary 2.5.7, we know T (R [X 1]) is von Neumann regular. Since R [X 1] is a PVMR and T (R [X 1]) is von Neumann regular, by

Theorem 2.7.1 we see that (R [X 1]) [X2] = R [X 1,X 2] is a PVMR. Further­ more by Corollary 2.5.7, we see that T (R [X 1,X 2]) is von Neumann regular.

By continuing in this fashion, we conclude that R [X 1, ... ,X n] is a PVMR.

Conversely suppose R [X 1, ... , Xn] is a PVMR for some arbitrarily fixed n G N. Then by Theorem 2.7.1 we know that R [X 1, ... ,X n-1] is a PVMR whose total quotient ring is von Neumann regular. By repeatedly applying

Theorem 2.7.1 in similar fashion, we find that each of the polynomial rings

32 3.1. WHEN IS R[Xx] A PVMR?

R[Xi,...,Xn-2], R[Xi,...,Xns],...,R[Xi,X2], and R[Xi] is a PVMR

whose total quotient ring is von Neuman regular. With one final appli­

cation of Theorem 2.7.1, we conclude that R is a PVMR and T(R ) is von

Neumann regular. ■

Lem m a 3.1.2. Let (XaIaga denote an arbitrary set of indeterminates.

Then R[Xa] is a PVMR if and only if R [Xi,... ,X n] is a PVMR for any finite subset of indeterminates { X 1, ... ,X n} C |Xa}.

Proof. Suppose that for all finite subsets of indeterminates { X 1, ... ,X n} C

Xa, R[Xi,...,Xn] = Rn is a PVMR and let A = (fi/gi,..., fk/gk) be

some finitely generated regular fractional ideal of R[Xa]. Note that each

generator of A is an element of T (R [X a]), and thus involves a finite num­

ber of the indeterminates from {X a}. Therefore there is some finite subset

of {X a}, say X A = { X 1, . . . , X m}, such that for all i E N

fi/gi E T(R[Xi,... ,Xm]). Let Am = (fi/gi,..., fk/gk)R[Xi,...,Xm].

Since R [X i , ... ,X m] is a PVMR by assumption, we know that Am is t-

invertible as it is finitely generated, fractional, and regular. Thus (AJ)* =

R [X i , ... , Xm] for some fractional ideal J of R [X i , ... , X m]. By Corollary

2.6.3, R[Xa] is a t-linked extension of R [X i , ... ,X m]. Thus since (A J)* =

R [X i,...,X m ], it follows that (AJR[Xa])* = R[Xa]. Let J' = JR[Xa].

33 CHAPTER 3. ARBITRARY POLYNOMIAL EXTENSIONS

As = R[X\], we see that A is a t-invertible ideal ofR[X\], and we conclude that R[X^] is a PVMR.

Conversely, suppose that R[Xx] is a PVMR, and consider the poly­ nomial subring R [X i,... ,X m] for some fixed subset ( X i , ... ,X m} C X a.

Let A = ( f 1/g1,..., f k/gk) be some finitely generated regular fractional ideal of R [X 1, . . . , X m]. Then A' = AR[X a] is a finitely generated reg­ ular fractional ideal ofR[X a]. Furthermore since R[X a] is a PVMR, we know that (A'J')* = R[X a] for some fractional ideal J' ofR[X a]. Let

J = J' n T (R [X 1, . . . , X m]), and for the sake of clarity denote Rm =

R [X 1, ... ,X m]. Observe that

Rm = (A'J')* n T(Rm) C (A'J' n T(Rm))* = (AJ)* C Rm

Thus (A J)* = R [X 1, . . . , X m]. Therefore we conclude that R [X 1, . . . , X m] is a PVMR. ■

Theorem 3.1.3. The ring R [X a] is a PVMR if and only if R is a PVMR and T(R) is von Neumann regular.

Proof. Suppose R[X a] is a PVMR. Then by Lemma 3.1.2, we know that for any finite set of indeterminates ( X 1, ... ,X n} CX a (with n G N fixed),

34 3.1. WHEN IS R[Xx] A PVMR? the polynomial ring R[Xi,... ,X n] is a PVMR. Thus by Corollary 3.1.1, we find that R is a PVMR and T(R ) is a von Neumann regular.

Suppose R is a PVMR and T(R ) is von Neumann regular, and let {X ^ j be an arbitrary set of indeterminates. Then by Corollary 3.1.1, for any fixed subset {Xl,..., X nj C X a, we know that R[Xl,..., X n] is a PVMR. Thus by Lemma 3.1.2, R[X\] is a PVMR. ■

35 CHAPTER 3. ARBITRARY POLYNOMIAL EXTENSIONS

3.2 W hen is R[X a] a Krull Ring?

Theorem 3.2.1. If R is a commutative ring, then R[Xx] is a KruH ring if and only if R is a finite direct product of KruH domains.

Proof. Suppose R is a finite direct product of Krull domains, say R =

Di X • • • X Dn for some fixed n G N, and let A be a regular ideal ofR[Xx].

Thus R[Xx] = Dl[Xx] X • • • Xn[X^]. D For the sake of clear notation, we denote D j[Xx] := D '. Since A is an ideal of a product of domains (or more generally, rings), observe that A = / l x • • • xIn, where L, is an ideal of D' for i = 1,... ,n. Specifically, we have

li = {x G D' : 3di G Di,..., dn G D^ s.t. (di,... ,x, di+i,..., dn) G A }.

Observe that if Ik is the zero ideal of theD'k for some fixed k G N

(A (Ifi x^^^x I„-i))t = R[Xx] therefore R[Xx] is a Krull ring.

Conversely suppose that R [Xx] is a Krull domain. It suffices to show that R [X ] is a Krull ring for some fixed X G {X x}. Fix such an X , and let

36 3.2. WHEN IS R[Xx] A KRULL RING?

A be a regular ideal ofR[X ]. By assumption, there is some fractional ideal

J of R[Xx] such that (A J)* = R[Xx]. Let J' = J n T (R [X ]). Note that J' is a fractional ideal.

Since A is t-invertible in R [Xx], we know that J = (R[Xx] : A). There­ fore J' = (R [X ] : A). Thus AJ' C R [X ], which implies (AJ')* C R [X ].

As R [Xx] = (AJ)*, we have R [X ] = (A J)* n T (R [X ]). Hence we have the following:

R[X] = (AJ)* n T(R[X]) C (AJ n T(R[X]))* = (AJ')*.

Thus R [X ] C (AJ')*. Therefore (AJ')* = R [X ], and thus R [X ] is a Krull ring. Therefore, by Theorem 2.7.2, we conclude that R is a finite product of Krull rings.

■

37 CHAPTER 3. ARBITRARY POLYNOMIAL EXTENSIONS

3.3 Future W ork

We conclude by giving some propositions and conjectures on UFRs of poly­ nomials, Dedekind polynomial rings, and the integral closedness of polyno­ mial rings. For the case of Dedekind rings, we have not yet formalized a complete proof. Though we can show the following propositions to be true.

Proposition 3.3.1. If R is a ring that is a finite product of Dedekind domains, then R[Xx] is a Dedekind ring.

Proof. Suppose R is a finite direct product of Dedekind domains, say R =

Di X ■ ■ ■ X D n. Then R[Xx] = Di[{Xx}] x ■ ■ ■ x Dn[{Xx}]. Let A be a regular ideal of R [Xx]. Then A = Ai x ■ ■ ■ x An, where each Aj is a nonzero ideal of D j[{X x}]. Furthermore, each Aj is invertible in D j[{X x}]. Letting

A = (A -1 X ■ ■ ■ X A - 1), we see that AA = R[Xx]. Therefore R[Xx] is a

Dedekind ring. ■

Proposition 3.3.2. If R[Xx] is a Dedekind ring, then R[X1,... ,X m] is a

Dedekind ring for any finite subset { X,... 1 ,Xm} ^ {X x}.

Proof. Let R be a commutative ring such that R [Xx] is a Dedekind ring.

Fix some arbitrary subset X ' = { X 1, . . . , X m} C {X x}, and let A be a

38 3.3. FUTURE WORK regular fractional ideal ofR[X^]. Let A = (R[Xx] : AR[Xx]) n T(R[X']).

Observe that

AA' = A((R[Xx] : AR[Xx]) n T(R[X'])) (3.1)

= A(R[Xx] : AR[Xx]) n T (R [X ']) (3.2)

= R[Xx] n T (R [X ']) (3.3)

= R[X']. (3.4)

Note that line 3.3 follows from the invertibility of A R[Xx] in the ring R[Xx].

Thus we have shown thatR[Xi,..., X m] is a Dedekind ring.

■

These two propositions point us in the direction of the following conjec­ ture.

Conjecture 3.3.3. If R is a commutative ring, then R [Xx] is a Dedekind ring if and only if R is a finite direct product of Dedekind domains.

For the case of UFRs, we have a similar conjecture.

Conjecture 3.3.4. If R is a commutative ring, then R [Xx] is a UFR if and only if R is a finite direct product of UFDs

39 CHAPTER 3. ARBITRARY POLYNOMIAL EXTENSIONS

We conclude with a conjecture that seems reasonable to make given the work done for the single-variable case in [10, 11]

Conjecture 3.3.5. If R is a ring, then R[X\] is integrally closed if and

only if R is integrally closed in T(R[X a ]) .

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42