On the Ideals and Semimodules of Commutative Semirings R. Olivier 2021 On the Ideals and Semimodules of Commutative Semirings By Ruan Olivier Submitted in fulfillment of the requirements for the degree of Master of Science (Mathematics) in the Faculty of Science at Nelson Mandela University April 2021 Supervisor: Dr S. Juglal Co-Supervisor: Mr Q.N. Petersen Declaration I, Ruan Olivier (215097939), hereby declare that this dissertation for Master of Science (Mathematics) is my own work and has not previously been submitted for assessment or completion of any postgraduate qualification to another university or for another quali- fication. Ruan Olivier PERMISSION TO SUBMIT FINAL COPIES OF TREATISE/DISSERTATION/THESIS TO THE EXAMINATION OFFICE Please type or complete in black ink FACULTY: Science SCHOOL/DEPARTMENT: Mathematics and Applied Mathematics I, (surname and initials of supervisor) Juglal S. and (surname and initials of co-supervisor) Petersen Q.N. the supervisor and co-supervisor respectively for (surname and initials of candidate) Olivier R. (student number) 215097939 a candidate for the (full description of qualification) Master of Science (Mathematics) with a treatise/dissertation/thesis entitled (full title of treatise/dissertation/thesis): On the Ideals and Semimodules of Commutative Semirings It is hereby certified that the proposed amendments to the treatise/dissertation/thesis have been effected and that permission is granted to the candidate to submit the final bound copies of his/her treatise/dissertation/thesis to the examination office. SUPERVISOR DATE And CO-SUPERVISOR DATE Acknowledgements Firstly, I am thankful to God for blessing me with a healthy mind and giving me the ability to complete my studies. I am grateful to my supervisors, Dr. Juglal and Mr. Petersen for their guidance, patience and the informative discussions we had. Additionally, I am indebted to my family and friends for all their emotional support and patience during my studies. In particular, I would like to thank my sister, Chant´ele Fourie. The discussions and moral support from her has played a significant role in helping me complete this dissertation. I am grateful to the friendly staff of the Department of Mathematics and Applied Mathematics for the role they played in my studies and growth as a person. In addition, I would like to convey my gratitude to the department for allowing me to lecture during 2020. Special thanks to Dr. Walton and Mr Petersen, whom I worked closely with during the entire 2020 academic year. I am eternally grateful for all the patience, advice and guidance you both gave me during this difficult year. Lastly, I am thankful for the financial support from the Nelson Mandela University and its Office of Research Development. Without the Vice Chancellor's scholarship and Postgraduate Research scholarship, my studies would not have been possible. Abstract Semirings are a generalisation of rings where additive inverses need not exist. In this dissertation, we focus on results of commutative semirings with non-zero identity. Many results that we study are analogous to results from commutative rings with non-zero identity. Properties which are unique to semirings are also investigated, such as semirings where all elements are additively idempotent. The notion of ideals is examined in the context of a semiring. Specifically, prime ideals, maximal ideals, k-ideals and partitioning ideals of semirings are considered. Additionally, the module over a ring is generalised to a semimodule over a semir- ing. The emphasis is on prime subsemimodules and multiplication semimodules. Lastly, invertible ideals of semirings are examined. Contents 1 Introduction1 2 Preliminaries3 3 Ideals 14 3.1 Prime Ideals.................................. 21 3.2 Maximal Ideals................................ 25 3.3 Principal Ideal Semirings........................... 29 3.4 Unique Factorisation Semidomains..................... 31 3.5 k-Ideals.................................... 33 3.6 Radical of Ideals and Primary Ideals.................... 36 3.7 Irreducible Ideals............................... 40 4 Constructing New Semirings from Existing Ones 43 4.1 Partitioning Ideals and Quotient Semirings................. 43 4.2 Total Quotient Semiring........................... 50 5 Semimodules 57 5.1 Invertible Ideals and Pr¨uferSemirings.................... 67 6 Gaussian Semirings 76 References 84 Chapter 1 Introduction A ring without additive inverses is called a semiring. In 1934, Vandiver [13] was first to explicitly introduce the concept of a semiring. However, semirings appear implicitly in the work of Dedekind as far back as 1894. Semirings have applications in many areas, including automata theory, combinatorics, cryptography and optimisation theory. [12] In this dissertation, we examine the properties of semirings. In particular, we focus on the ideals of semirings and semimodules over semirings. The results that we consider are based on existing results for rings. We will assume that a semiring contains a multiplicative identity such that it does not coincide with the additive identity. Furthermore, we will only consider semirings with commutative multiplication. In chapter 2 we gather all the preliminary definitions, terminology and results that we will use throughout this dissertation. Additionally, examples of semirings are provided. We consider different types of ideals of a semiring in chapter 3. Firstly, we list prop- erties that the operations on ideals have. Then we consider prime ideals and maximal ideals. Most of the results that appear here are the same as for rings. In the prime ideals section we prove the prime avoidance lemma for semirings. Next, we investigate results on principal ideal semirings and unique factorisation semidomains which are also similar to their ring counterparts. In the following section, we focus on k-ideals. Finally, we consider primary ideals and irreducible ideals, which are used to prove that every k-ideal in a Noetherian semiring has a primary decomposition. Chapter 4 illustrates how new semirings can be constructed from existing semirings. We start by introducing partitioning ideals which form a quotient semiring. Next, we demonstrate how the total quotient semiring is formed. We investigate the ideals of both quotient semirings and the total quotient semiring. A module over a ring is generalised to semimodules over a semiring in chapter 5. We prove results on prime subsemimodules, before investigating multiplicative semimodules. The rest of chapter 5 is dedicated to invertible ideals and Pr¨ufersemirings. For any 1 CHAPTER 1. INTRODUCTION 2 semiring, an invertible ideal is finitely generated. A Pr¨ufersemiring is a semiring such that every finitely generated ideal is invertible. Lastly, we provide and prove a list of statements which are equivalent to a semiring being Pr¨ufer. Finally, in chapter 6 we examine Gaussian semirings. These semirings satisfy the content formula c(f)c(g) = c(fg) for all polynomials f and g over the semiring. The Dedekind-Mertens lemma for semimodules is proved. It is shown that every Pr¨ufersemir- ing is a Gaussian semiring. Next, we prove the Gilmer-Tsang theorem for semirings, which provides sufficient conditions for a Gaussian semiring to be a Pr¨ufersemiring. The disser- tation is concluded by showing that the semiring of finitely generated ideals of a Pr¨ufer semiring is both a Pr¨uferand Gaussian semiring. Chapter 2 Preliminaries Definition 2.1 [12] A nonempty set M along with an associative binary operation ∗ defined on M is called a semigroup (M; ∗). Definition 2.2 [12]A monoid (M; ∗) is a semigroup which also has an identity element e. That is, there exists an e 2 M such that m ∗ e = m = e ∗ m for all m 2 M. Definition 2.3 [14] A monoid (G; ∗) is called a group if for every nonzero element g 2 G there exists an element h 2 G such that g ∗ h = e = h ∗ g. If G is commutative with respect to ∗, then G is called an abelian group. Definition 2.4 [14]A ring (R; +; ·) is a nonempty set R on which the binary operations of addition and multiplication have been defined such that the following are satisfied: (i)( R; +) is an abelian group. (ii)( R; ·) is a semigroup. (iii) Multiplication distributes over addition from either side. If (R; ·) is instead a monoid with identity element 1, then we say R is a ring with identity. The ring R is said to be commutative if multiplication is commutative. Definition 2.5 [12]A semiring (S; +; ·) is a nonempty set S on which the binary op- erations of addition and multiplication have been defined such that the following are satisfied: (i)( S; +) is a commutative monoid with identity element 0. (ii)( S; ·) is a monoid with identity element 1. (iii) Multiplication distributes over addition from either side. (iv) The element 0 is the absorbing element of multiplication. That is, 0 · s = 0 = s · 0 for all s 2 S. 3 CHAPTER 2. PRELIMINARIES 4 The semiring S is said to be commutative if multiplication is commutative. Remark 2.6 In the case when 1 = 0, we have that s = s · 1 = s · 0 = 0 for all s 2 S. Hence S = f0g. We want to avoid the trivial case when S = f0g and will therefore assume that 1 6= 0. Remark 2.7 From this point onward we will only consider commutative semirings and will not make specific mention of it. Definition 2.8 [12] A nonempty subset T of a semiring S is a subsemiring of S if T itself is a semiring. More specifically, T is a subsemiring of S if 0; 1 2 T and T is closed under addition and multiplication. Example 2.9 Any ring with identity is a semiring. Now, consider the set of all non- + + negative integers Z0 together with the usual addition and multiplication. Then (Z0 ; +; ·) + + is a semiring, but not a ring. Similarly, the sets Q0 and R0 together with the usual + + + + operations are semirings too. Clearly Z0 is a subsemiring of Q0 , while Z0 and Q0 + are subsemirings of R0 .