Unique Factorisation in Abstract Algebra Adrian Petersen Supervised by Phill Schultz University of Western Australia 1 Abstract The Fundamental Theorem of Arithmetic states that any integer can be written as the product of prime numbers and units. This decomposition is unique up to order and unit multiples. This familiar factorisation of integers can be generalised to more abstract objects such as rings, and sets of ideals or modules. The factorisation properties of each of these is investigated. Further, definitions and properties of Factorial Domains, Dedekind Domains, Factorial Monoids and Krull{Schmidt classes of modules are outlined. An effort is then made to relate each of these to a classic example of a factorial monoid, the (Ω) free commutative monoid N . We have confirmed that an isomorphism can be constructed from either a Factorial Domain, set of ideals of a Dedekind Domain, and Krull{Schmidt class of modules to this free commutative monoid (Ω) N . 1 Introduction The Fundamental Theorem of Arithmetic states that every non-zero integer can be expressed uniquely as a product of prime numbers and units. This theorem is com- monly taught in an introduction course to pure mathematics. The aim of this project is to investigate similar factorisation behaviour in the more abstract settings of rings, monoids, ideals and modules. First we shall be considering the factorisation of ele- ments in rings and monoids. Then we consider the factorisation of sets such as ideals and modules. Common features between the decomposition of these objects will be investigated and put into a similar framework. The reader is assumed to be familiar with concepts from an undergraduate algebra course, i.e. they are familiar with the concepts of an integral domain, multiplicative monoid, ideal of a ring and module over a ring. R is an integral domain with unit group R×. N is the natural numbers, N+ is the set of positive natural numbers, and Z represents the integers. Section two serves as an introduction to unique factorisation in a more abstract setting, but goes no further than an undergraduate algebra course. Section three introduces unique factorisation of elements in a monoid, and relates it to section two by means of an isomorphism to the set of functions f :Ω ! N. It is then found similar isomorphisms can be constructed between sets of ideals, and modules with unique factorisation properties. 2 Factorial Domains The purpose of this section is to introduce the idea of factorisation in rings. The concept of factorisation in a ring is easiest to characterise if the ring R is a domain which we shall assume from now on. Familiar examples will be discussed in this sec- tion from number theory. The idea of a Unique Factorisation Domain is analogous to the Fundamental Theorem of Arithmetic for the integers which states that every non{zero integer can be uniquely expressed as the product of prime numbers and units. 2.1 Elementary Definitions and examples Well known examples of integral domains, which we shall call simply domains, are the integers Z, the fields Zp of integers modulo a prime p, any subring containing 1 of a field F, the polynomial rings R[x1; : : : ; xn] in several variables over a domain R. 1 Definition 2.1.1. Let R be a domain. Then: (a)a unit is an invertible element of R. Let R× denote the group of units in R. a; b 2 R are said to be associates if a = bu for some u 2 R×. (b) Let a 2 R, if a = bc, then b and c are factors of a. An atom of R is a non-zero non-unit such that if a = bc then b or c is a unit. A domain is said to be atomic if every element, not zero or a unit , is a product of a finite set of atoms. (c) Let a; b 2 R. Then a is said to divide b , denoted a j b if and only if b = ac for some c 2 R. p 2 R is said to be prime if p j ab implies p j a or p j b 8 a; b 2 R. Example 2.1.1. Examples of atomic domains include (a) Z p (b) Z[ −5] (c) R[x; y] where R is a domain If R is not atomic then there must exist a 2 R such that if a is a product of finitely many elements, they cannot all be atoms. For an example of a domain that is not atomic consider the ring R = ff : C ! Cg where C is the complex numbers, addition and and multiplication are pointwise. Then for any f 2 R, and n 2 N you can factorise f = gn where g(z) is an n-th root of f(z). So R has no atoms. 2.2 Factorisation into Atoms Definition 2.2.1. An element a 2 R has a unique decomposition into atoms up to order and associates if whenever a = b1 : : : bn = c1 : : : ck for atoms bi; ci then n = k and for each i 2 [1; n] there exists j 2 [1; n] such that ai and bj are associates. Factorisationp into atomsp in atomic domains need not be unique. Consider the domain [ −5] = a + b −5 j a; b 2 . It is possible to show that the element pZ Z 6 2 Z[ −5]. 6 can be factorised into two distinct products of atoms. p p 6 = 2 · 3 = (1 + −5)(1 − −5) p p p It is simple to prove that 2,3,(1 + −5) and (1 − −5) are atoms in Z[ −5], none of which are associates of each other. 2 p Proof.p Suppose that 2 can be written as a product of non-units 2 = (a + bp−5)(c + dp−5). Taking the norm of each side of this equation gives j2j = ja + b −5jjc + 2 2 2 2 2 2 d −5j which simplifies to 4 = ja + 5b jjc + 5d j. So ja + 5bpj = 1; 2 or 4,p which has integral solutions (a; b) = (1; 0) or (2; 0). Hence either (a+b −5) or (c+d −5) is a unit. p p Similarly, it can be shown that 3,(1+ −5) and (1− −5) can only be factorised into a product of units and themselves. p Each of these is an atom, the only units in Z[ −5] are 1 and −1, so obviously none of these atoms are associates. 2.3 Prime Factorisation in Domains Let a 2 R where R is a domain, a = p1::::pn is called an prime factorisation of a if p1; :::; pn are all primes. Lemma 2.3.1. Let R be a domain and let p 2 R. If p is prime then p is an atom. Proof. Suppose that p 2 R is prime, by definition it is not a unit. Now if p = ab for some a; b 2 R, then p j a or p j b. Without loss of generality suppose that p j a, then a = pc for some c 2 R. Then p = ab = pcb, since R is a domain the cancellation law holds so p = pcb implies that 1 = cb. So b is a unit. Similarly it can be shown that if p j b then a is a unit. Beacause p is not a unit and cannot be expressed as a product of non-units, p is an atom. However, the converse of this statement is not always true. An atom in a domain need not be prime.p An examplep is the bestp way to illustrate this. Ifp we againp consider the atoms a = 1+ −5; b = 1− −5 in Z[ −5], note that ab = (1+ −5)(1− −5) = −4. So ab j −4, but a - −4 and b - −4 so neither a or b are prime. In section 2.2. we showed by counter example that factorisation into atoms need not be unique in domains. When considering primes, it turns out that factorisation is unique up to order and unit multiple. Lemma 2.3.2. In any domain if an element a = p1 : : : pn = q1 : : : qm where each pi and each qj are primes, then n = m and for each i there exists a unique j such that pi and qj are associates. Proof. Suppose a = p1 : : : pn = q1 : : : qm where 1 ≤ k ≤ n for primes pi and qj.I prove by induction on n that k = n and each qj is an associate of some pi. If n = 1, then p1 = q1, so the statement is true for n = 1. Assume the statement is true for all m ≤ n. Suppose a = p1 : : : pn = q1 : : : qk where k ≤ n for primes pi and qj. Then 3 consider p1, note that p1 j q1 : : : qk so p1 j qj for some j 2 [1; k] because p1 is prime. Since p1 and qj are prime, they are atoms, so p1 is associate to qj. Hence up1 = qj × for some u 2 R . Relabelling qj to q1 gives us p1 : : : pn = q1 : : : qk = up1q2 : : : qk. Cancelling the p1 from each side of the equation leaves us with p2 : : : pn = uq2 : : : qk. By the induction assumption, n − 1 = m − 1 and each qj is an associate of some pi. Hence the stement is true for n. The Fundamental Theorem of Arithmetic provides a nice example of this in the integers Z. In the domain of integers every atom is a prime. It is natural to now ask the question, when are primes and atoms the same thing, and when is a factorisation into atoms unique.