Weakly Integrally Closed Domains and Forbidden Patterns by Mary E. Hopkins A Dissertation Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Ful…llment of the Requirements for the Degree of Doctor of Philosophy Florida Atlantic University Boca Raton, Florida May 2009 Acknowledgements I am deeply grateful to my advisor, Dr. Fred Richman, for his patience and gentle guidance, and for helping me to understand algebra on a deeper level. I would also like to thank my family and Dr. Lee Klingler for supporting me personally through this challenging process. Many thanks are given to Dr. Timothy Ford and Dr. Jorge Viola-Prioli for their insightful remarks which proved to be very helpful in writing my dissertation. Lastly, I will be forever grateful to Dr. James Brewer for taking me under his wing, opening my eyes to the beautiful world of algebra, and treating me like a daughter. I dedicate this dissertation to my grandmother and Dr. Brewer. iii Abstract Author: Mary E. Hopkins Title: Weakly Integrally Closed Domains and Forbidden Patterns Institution: Florida Atlantic University Dissertation advisor: Dr. Fred Richman Degree: Doctor of Philosophy Year: 2009 An integral domain D is weakly integrally closed if whenever there is an element x in the quotient …eld of D and a nonzero …nitely generated ideal J of D such that xJ J 2, then x is in D. We de…ne weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of …nitely many 0’s and 1’s is forbidden if whenever the characteristic binary string of a numerical monoid M contains F , then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every …nite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed. iv Contents 1 Introduction .................................................................1 2 Background and Preliminary Results .....................................4 3 Forbidden Patterns .........................................................9 3.1 Terminology and Preliminary Results . 9 3.2 Examples of Forbidden Patterns . 12 3.3 A Characterization of Forbidden Patterns . 15 4. A Test to Determine if a Monoid is Weakly Integrally Closed . 20 4.1 An Algorithm to Determine if a Monoid has a Bad Zero . 20 4.2 A Characterization of Weakly Integrally Closed Monoids. .23 5 Question 1 Generalized ....................................................25 6 Weakly Integrally Closed Domains .......................................30 7 Open Problems.............................................................37 References .....................................................................39 v 1 Introduction Noether introduced the concept of integrality in the 1920s. Let D D0 be integral n n 1 domains. An element x D0 is said to be integral over D if x + dn 1x + + 2 d1x + d0 = 0 for some natural number n and di D. Kaplansky cites an equivalent 2 de…nition in [6]: x D0 is integral over D if there is a …nitely generated D-submodule 2 J of D0 such that xJ J. An element x D0 is almost integral over D if there is 2 n a nonzero d D such that dx D for every natural number n. Any element x D0 2 2 2 that is integral over D is also almost integral over D. We say that the domain D is (completely) integrally closed in D0 if whenever an element x D0 is (almost) 2 integral over D, then x is in D. In [3], Brown gives an example of a 3-by-3 matrix over the ring D = K[[t3; t4]], where K is a …eld and t an indeterminate, whose minimum polynomial, X2 t5X, has coe¢ cients which do not lie in D. Brewer and Richman, intrigued by Brown’s example, ask when must the coe¢ cients of the minimum polynomial of a n-by-n matrix over a domain lie in that domain. In [2], this question led to the notion of a weakly integrally closed domain. An element x D0 is said to be strongly integral 2 over D if there is a nonzero …nitely generated ideal J of D such that xJ J 2. If we use Kaplansky’s de…nition of an integral element, then it is easy to see that any element x D0 that is strongly integral over D is also integral over D. We say that D 2 is weakly integrally closed in D0 if every element x D0 that is strongly integral 2 over D is in D. If D is weakly integrally closed in it’squotient …eld, then we simply say that D is weakly integrally closed. Let M be a numerical monoid (i.e. an additive submonoid of the natural numbers) 1 m1 m2 and let K[M] = k0 + k1t + k2t + ::: : ki K and mi M , where K is a …eld. f 2 2 g We represent M by its in…nite characteristic binary string. For example, the string 10111 ::: represents the monoid generated by 2 and 3. Brewer and Richman show the following important results: Theorem 1 ([2], Theorems 1 and 3) If the binary string of a monoid M contains a stretch of the pattern 11011, then K[M] is not weakly integrally closed. For example, the monoids M = 0; 3; 4; 6; 7; 8 ::: and M 0 = 0; 6; 8; 12; 14; 15; 16;::: f g f g have the following in…nite binary string representations: M : 10011011111 ::: and (1) M 0 : 100000101000101111 ::: . (2) Since the binary strings of M and M 0 contain the patterns 11011 and 1 1 0 1 1, respectively, then the rings K [M] and K [M 0] are not weakly integrally closed. Thus, Brown’sring K[t3; t4] is not weakly integrally closed. Theorem 2 ([2], Theorem 5) Let i and m be a pair of natural numbers such that m 2i and let M = 0; i; m; m + 1;::: . Then K[M] is weakly integrally closed. f g For example, the ring K[t4; t7; t8;:::] is weakly integrally closed. The motivation for this dissertation is to answer the following two questions asked by Brewer and Richman in [2]. Question 1 Are weakly integrally closed monoid rings K [M] characterized by the property that the binary string of the monoid M contains no stretch of the pattern 11011? Question 2 Is the ring K[t4; t5; t11] weakly integrally closed? 2 When investigating a particular property of monoid rings, it is common for math- ematicians to study the analogous property in the monoid itself. In Gilmer’s book, Commutative Semigroup Rings, he extends the concept of integrality to monoids and studies the relationship between integrally closed/completely integrally closed monoids and their corresponding monoid rings. Similarly, Geroldinger has exten- sively researched completely integrally closed monoids and their rings in papers such as The Complete Integral Closure of Monoids and Domains [4]. We will do the same by de…ning strongly integral elements for monoids. In doing so, we will negatively answer Question 1 and a¢ rmatively answer Question 2. In Chapter 2, the concept of being weakly integrally closed is extended to monoids and it is shown that if a monoid ring is weakly integrally closed, then so is the monoid. In Chapter 3, we give examples of patterns of …nitely many 0’sand 1’ssuch that when found in the characteristic binary string of a numerical monoid, then the monoid ring is not weakly integrally closed; we call such patterns forbidden. Question 1 is answered in Example 31. In Chapter 4, we discuss a JavaScript program that we wrote which determines if a pattern is forbidden. We prove that this program determines whether or not a numerical monoid is weakly integrally closed. Chapter 5 is devoted to showing that there is no …nite set S of forbidden patterns such that whenever a monoid M contains no stretch of a pattern in S, then M is weakly integrally closed. In Chapter 6, we show that particular monoid rings are weakly integrally closed. We answer Question 2 and show that the ring K tl; tl+1; tl+s; tl+s+1;::: is weakly integrally closed for all 3 < s l. 3 2 Background and Preliminary Results We list some relevant notation, de…nitions and results used throughout the rest of this dissertation. Throughout this dissertation, all rings are commutative and contain 1. Let D D0 be domains. De…nition 3 ([2]) An element x in D0 is said to be strongly integral over D if there exists a nonzero …nitely generated ideal J of D such that xJ J 2. De…nition 4 ([2]) The domain D is weakly integrally closed in D0 if every ele- ment x D0 that is strongly integral over D is an element of D. 2 Theorem 5 ([2], Theorem 2) 1. (Intersections) If each member in a family of subrings Di of D is weakly inte- grally closed in D, then iDi is weakly integrally closed in D. \ 2. (Transitivity) If D is weakly integrally closed in D0, and D0 is weakly integrally closed in D00, then D is weakly integrally closed in D00. 3. (Weakness) If D is integrally closed in D0, then D is weakly integrally closed in D0. De…nition 6 A monoid is a nonempty set closed under an associative binary oper- ation that has an identity element. For the rest of this dissertation, we assume that all monoids are commutative and we denote the monoid operation by +.