Commutative Hyperalgebra

Commutative Hyperalgebra A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematical Sciences of the McMicken College of Arts and Sciences by Nadesan Ramaruban B.Sc. University of Jaffna, November 2003 December, 2013 Committee Chair: Tara Smith, Ph.D. Abstract This work develops the notion of hyperrings as an extension of classical ring theory. Rings are sets that possess algebraic operations of addition and multiplication of elements in the set, encompassing examples such as the integers, rational numbers, etc. Hyperrings generalize rings further to allow multiple output values for the addition operation. In mathematics, introducing more abstract definitions often serves to clarify and consolidate phenomena that arise in a variety of settings. Hyperrings arise naturally in several settings in algebra, including quadratic form theory, number theory, order- ings and ordered algebraic structures, tropical geometry, and multiplicative subgroups of fields. In this work we present an overview of the history of the development of the theory of hyperrings and other algebraic objects possessing hyperoperations. We continue by presenting a number of illustrative examples of hyperrings. We then de- termine the extent to which many results of classical commutative ring theory can be generalized to the setting of hyperrings. This in turn sheds light on the specific re- alizations of these objects. In particular, we consider homomorphisms of hyperrings, develop the hyperideal theory for hyperrings, introduce hypermodules and hypervector spaces, and examine the structure of specific classes of hyperrings such as Artinian and Noetherian hyperrings, Euclidean, principal hyperideal, and unique factorization hyperdomains, and hyperfields. i ii Acknowledgements I would like to express my deep sense of gratitude and appreciation to all of the following people for their support and encouragement throughout this journey. To Dr Osterburg, for having been my teacher of algebra throughout the algebra courses and for serving in my dissertation committee; to Dr Nageswari, without whom this journey wouldn’t have started; to my adviser Dr Smith, for her friendly guidance and patience over the last few years; to Dr Hodges, for serving in my dissertation committee. Lastly, to my parents and brother, who have always stood beside me, and sacrificed much to have made this journey possible iii Contents Abstract i Acknowledgements iii 1 Introduction 1 2 Examples 6 2.1 Explicit small examples . 7 2.2 Krasner’s construction and a counter example . 11 2.3 Examples from Real Algebra . 12 2.4 Examples from Quadratic Form Theory . 14 2.5 Hyperfields arising from linearly ordered groups . 15 2.6 Examples from Tropical Geometry and Number Theory . 16 2.7 Some Ad Hoc Examples . 18 3 Hyperrings 20 4 Operations on hyperideals 31 5 Hyperdomains 45 5.1 Euclidean Hyperdomains . 45 5.2 Principal Hyperideal Hyperdomains and Unique Factorization Hyper- domains . 50 6 Hypermodules 55 7 Hyperrings and Hypermodules of Fractions 63 iv 7.1 The construction of hypermodules of fractions . 66 8 Primary Decomposition 75 9 Chain Conditions 81 10 Hypervector Spaces 86 11 Noetherian Hyperrings 92 12 Artinian Hyperrings 96 13 Discrete Hypervaluation Hyperrings and Dedekind Hyperdomains 100 14 Some Hyperfield Theory 111 15 Conclusion 114 References 117 v 1 Introduction Even though the concept of hyperstructures was introduced in the early part of the twen- tieth century, it remained largely unappreciated until the late 70’s. Now, with time, it has grown into a separate branch of mathematics having many applications in other areas such as theory of automata and languages, particle physics[8] and artificial intelligence, and possessing rich avenues for further research. In 1934, F. Marty [16] introduced the concept of a hyperoperation and thereby defined an algebraic structure called a hypergroup. The notion of a hyperoperation is a straight- forward generalization of the notion of a binary operation. Specifically, the output of a hyperoperation is allowed to be a non-empty set whereas the output of a binary operation is always a single element. According to Marty, the definition of a hypergroup is as follows: Let H be a nonempty set, ◦ be a hyperoperation satisfying 1. a ◦ (b ◦ c) = (a ◦ b) ◦ c for all a; b; c 2 H 2. a ◦ H = H ◦ a for all a 2 H S S then (H; ◦) is a hypergroup. Here a ◦ (b ◦ c) = x2(b◦c) a ◦ x and a ◦ H = x2H a ◦ x and so on. In 1956, Marc Krasner [12] introduced the concepts of hyperrings and hyperfields. He extracted the additive structure of a hyperring as “canonical hypergroup”. This same notion was introduced independently by M. Marshall, in 2006, with the name “multigroup”[14]. According to Krasner, a canonical hypergroup H is a nonempty set equipped with a hyperoperation + satisfying the following: 1 1. (a + b) + c = a + (b + c) for all a; b; c 2 H. 2. a + b = b + a for all a; b 2 H. 3. 9 0 2 H such that a + 0 = fag. 4. 8a 2 H; 9 − a 2 H such that 0 2 a + (−a) = (−a) + a. This specializes Marty’s definition of hypergroup by requiring commutativity and ex- istence of identity and inverses, making the definition a generalization of the familiar abelian group structure. From here onward, in this work, whenever the term “hypergroup”is used it only refers to “canonical hypergroup”in the sense of Krasner. Krasner introduced the concept of hyperrings and hyperfields as a technical tool in a study of approximations of complete valued fields by a sequence of such fields. Accord- ing to Krasner, the formal definition of hyperrring is as follows: A hyperring (R; +; ·) is an algebraic structure having the following properties: 1. (R; +) is a canonical hypergroup 2. (R; ·) is a semigroup with identity having 0 as bilaterally absorbing element. That is a · 0 = 0 · a = 0 for all a 2 R 3. The multiplication · is distributive. That is a · (b + c) = a · b + a · c (b + c) · a = b · a + c · a for all a; b; c 2 R. In a hyperring with multiplicative identity 1, if every nonzero element is invertible then it is called a hyperfield. 2 The structure of these objects are perhaps most easily understood by considering a motivating example that arises from observing the signs of numbers in usual operations. When +1 represents “positive”and −1 represents the “negative”and 0 represents the “zero” in real numbers, we have an easily understood hyperfield of three elements, namely Q = {−1; 0; 1g. Here addition and multiplication are defined in the obvious way as follows: + 0 1 -1 · 0 1 -1 0 0 1 -1 0 0 0 0 1 1 1 Q 1 0 1 -1 -1 -1 Q -1 -1 0 -1 1 Then (Q; +; ·) forms a hyperfield in the sense of Krasner. During the 80’s two other notions of hyperrings were introduced. One is a hyperring in which both addition and multiplication are hyperoperations. This kind of hyperring was studied by DeSalvo and Barghi [2][6] and by A. Asokkumar and Velrajan[24]. Another kind of hyperring is one in which addition is a binary operation but multiplication is a hyperoperation. Rota [21] introduced this hyperring and did extensive study on this. Our work exclusively focuses on hyperrings in the sense of Krasner. These hyperrings are often referred to as Krasner hyperrings. In this work, from now on, hyperrings will always be taken to to mean Krasner hyperrings unless otherwise specified. There is another kind of hyperstructure called “multirings”which was introduced by 3 M. Marshall[14]. Multirings naturally arise in the study of spaces of signs, also known as abstract real algebra. Objects which arise in the study of constructible sets in real geometry are also shown to be multirings. The only difference between hyperrings and multirings is that hyperrings have the strong distributive property whereas multirings have the weak distributive property. Specifi- cally, if a; b; c are in a hyperring (R; +; ·), we have a · (b + c) = a · b + a · c. If a; b; c are in a multiring (R; +; ·), we have a · (b + c) ⊆ a · b + a · c. If each element in a multiring is invertible then it is called multifield. The notions hyperfield and multifield are exactly the same. For if a(b + c) ( ab + ac then a−1(a(b + c)) ( a−1ab + a−1ac. So b + c , b + c. In recent times, many results have been published on hyperrings, most of which were ex- tensions of the standard results in commutative algebra. Notably, a paper by B.Davvaz (2003) [5] extended the isomorphism theorems, another paper by Davvaz and Salasi (2006)[4] extended the Chinese Remainder Theorem to hyperrings. These authors also defined the concepts of localization and hypervaluation in hyperrings, stating and prov- ing certain analogous results relating to these concepts. In 2011, Asokkumar and Vel- rajan [24] proved isomorphism theorems for hyperrings with two hyperoperations, and also proved that in order to define a quotient hyperring, the hyperideal concerned need not be normal. That is, if I is hyperideal of a hyperring R we don’t need the property that x − x ⊆ I for all x 2 R to define R=I, which was a necessary condition for the statements and proofs of isomorphism theorems for Krasner hyperrings, according to Davvaz and Salasi. Papers by B.Davvaz and Salasi, Asokkumar and Velrajan provided the inspiration to 4 look for other important results from commutative algebra that can be transferred to hyperrings. In this work, we define hyperdomains and prove certain results regarding principal hyperideal hyperdomains, euclidean hyperdomains and unique factorization hyperdomains.

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