Unit 14 Integral Domains and Fields UNIT 14 INTEGRAL DOMAINS AND FIELDS Structure Page Nos. 14.1 Introduction 125 Objectives 14.2 What is an Integral Domain? 126 14.3 Characteristic of a Ring 132 14.4 Fields 135 14.5 Field of Quotients 139 14.6 Prime and Maximal Ideals 143 14.7 Summary 150 14.8 Solutions / Answers 151 14.1 INTRODUCTION In Unit 10, we introduced you to rings, and then to special rings, like commutative rings and rings with unity. As you found there, the speciality of these rings lies in the properties of the multiplication defined on them. You also saw that a typical example of such special rings is Z. So, in a sense, these rings are abstractions of Z. Yet, they do not necessarily satisfy an essential property of Z, which is the cancellation property for multiplication. In this unit, you shall study about rings which have this property too. Such rings are called integral domains, and are very important for studying several branches of algebra and its applications. Throughout this unit, we shall assume the rings to be commutative, unless specified otherwise. In Sec.14.2, we will begin by discussing what a zero divisor is. This will take you further, to the definition of an integral domain, along with several examples. You will see why many of the rings you have seen so far are examples of integral domains, and why many are not! We will discuss various properties of integral domains also in this section. In the next section, Sec.14.3, we will focus on a feature that characterises any ring, not necessarily commutative. This is a non-negative integer connected to each ring, called its characteristic. We will focus here on the characteristic of an integral domain, in particular. You will study the reasons for the characteristic of an integral domain being 0 or a prime number. 125 Block 4 Integral Domains In Sec.14.4, you will study a common property of rings like Q, R, C and Zp (where p is a prime number). In these rings, the non-zero elements form an abelian group with respect to multiplication. Such a ring is called a field. Fields are very useful algebraic objects, one reason being that every non-zero element of a field is a unit. In this section, you will also study some basic properties of fields. Next, in Sec.14.5, you will study that given an integral domain, there is a field containing it. You will also see how to construct the smallest field that contains a given integral domain. As you will see, this is essentially the way that Q is constructed from Z. Related to integral domains and fields are certain special ideals of rings, called prime ideals and maximal ideals. In Sec.14.6, you will study such ideals and their relationship with integral domains and fields. As you can see, in this unit you will study several new concepts. You may need some time to grasp them. Don’t worry. Take as much time as you need. But by the time you finish studying it, we hope that you will have attained the following learning objectives. Objectives After studying this unit, you should be able to: • define, and give examples of, a zero divisor in a ring; • check whether an algebraic system is an integral domain or not; • obtain the characteristic of a ring, whether commutative or not; • check whether an algebraic system is a field or not; • prove, and apply, simple properties of integral domains and fields; • construct, or identify, the field of quotients of an integral domain; • define, and identify, prime ideals and maximal ideals of a ring. 14.2 WHAT IS AN INTEGRAL DOMAIN? Let’s begin this discussion with looking at the product of two non-zero integers. You know that this is a non-zero integer, i.e., if m, n ∈Z such that m ≠ 0, n ≠ 0, then mn ≠ 0. Now consider the ring Z6. Here 2 ≠ 0 and 3 ≠ 0, yet 2 ⋅ 3 = 6 = 0. So, we find that the product of the non-zero elements 2 and 3 is zero in Z6. This example leads us to the following definition. Remember that R is Definition: A non-zero element r in a ring R is called a zero divisor in R if commutative. there exists a non-zero element b in R such that rb = 0. (Note that b will be a zero divisor too!) Now, do you agree that 2 is a zero divisor in Z6? What about 3 in Z4? Since 3 ⋅ x ≠ 0 for every non-zero x in Z4 , 3 is not a zero divisor in Z4. The name ‘zero divisor’ comes from the fact that an element x ∈R divides r ∈R if ∃ y∈R s.t. xy = r. The difference, though, is that here r = 0 but x 126 Unit 14 Integral Domains and Fields and y are both non-zero. So, in the case of 2 ⋅ 3 = 0, 2 0 and 3 0. Thus, both 2 and 3 are zero divisors in Z6 . Let us consider some more examples of rings with zero divisors. Example 1: Check whether or not Z[ 3] < 4 > has zero divisors. Solution: Note that Z[ 3] < 4 > ={a + b 3+ < 4 > a, b∈Z}, and < 4 > = {4a + 4b 3 a, b∈Z}. We need to see if ∃ x, y∈Z[ 3] < 4 > s.t. x ≠ 0, y ≠ 0 but xy = 0, i.e., if ∃ x, y∈Z[ 3] s.t. x, y∉< 4 >, but xy∈< 4 > . Consider x = 2, y = 2. Now x ∉< 4 >, y ∉< 4 >. (Why?) But xy = 4∈< 4 > . Hence, 2 is a zero divisor in Z[ 3] < 4 >. You can find several other zero divisors in this ring. In fact, you should try and find at least one more. *** Example 2: Give an example, with justification, of a zero divisor in C[0,1]. Solution: Consider the function f ∈C[0,1], given by ⎧ ⎪ x − 1 , 0 ≤ x ≤ 1/ 2 f(x) = ⎨ 2 ⎩⎪ 0, 1/ 2 ≤ x ≤ 1. Let us define g :[0,1] → R by 0, 0 ≤ x ≤ 1/ 2 ⎪⎧ g(x) = ⎨x − 1 , 1/ 2 ≤ x ≤ 1. ⎩⎪ 2 Then, from Calculus, you know that f , g ∈C[0,1]. Also f ≠ 0, g ≠ 0 and (fg)(x) = f (x)g(x) = 0 ∀ x ∈[0, 1]. Thus, fg is the zero function. Hence, f is a zero divisor in C[0,1], and so is g. *** Example 3: Check whether or not the direct product of two non-trivial rings has zero divisors. Solution: Let A and B be non-trivial rings. Let a ∈A, a ≠ 0, and b∈ B, b ≠ 0. Then (a,0)∈A× B and (0,b)∈A × B are both non-zero. However, (a,0)(0,b) = (0,0). Hence, (a,0) and (0, b) are zero divisors in A × B. *** Example 4: Check whether or not ℘(X) has zero divisors, where X is a set with at least two elements. 127 Block 4 Integral Domains Solution: Each non-empty proper subset A of X is a zero divisor because A ⋅Ac = A ∩ Ac = «, the zero element of ℘(X). *** Example 5: Let x be a zero divisor in Z n , where n ∈N. Show that (x, n) >1. Solution: Let y∈Z n s.t. y ≠ 0 and x y = 0. So n x, n y, but n xy. Suppose (x, n) =1. Then ∃ m,r ∈Z s.t. mx + nr = 1. Then y = y⋅1 = mxy + nry. Now n xy and n nry. Hence, n (mxy + nry), i.e., n y, a contradiction. Thus, (x, n) ≠ 1. Hence, (x, n) > 1. *** Try solving these exercises now. E1) List all the zero divisors and all the units in Z, and in Z10. Is there a relationship between the zero divisors and the units of each ring? If so, what is it? (This is linked with E5.) E2) Prove the converse of what is given in Example 5. E3) Let R be a ring and a ∈ R be a zero divisor. Show that every non-zero element of the principal ideal Ra is a zero divisor. E4) Check whether or not ℘(X) has zero divisors, where X = {a}. E5) Let R be a ring with unity and a ∈ R, a ≠ 0. i) If a is not a zero divisor, does a∈U(R)? ii) If a∈U(R), can a be a zero divisor in R? Justify your answers. So far you have seen several examples of rings with zero divisors. You also know that Z has no zero divisors. Actually, there are many rings without zero divisors. Let us define such rings. Definition: A non-trivial ring R is called an integral domain if i) R is commutative, ii) R is with identity, and ii) R has no zero divisors. Thus, an integral domain is a non-trivial commutative ring with identity in which the product of two non-zero elements is a non-zero element. This kind of ring gets its name from the set of integers, one of its best known examples. In fact, integral domains were originally thought of as a 128 generalisation of Z. Unit 14 Integral Domains and Fields Can you think of other integral domains? What about Q, R and C? You should check that they satisfy the conditions in the definition. Now, from the examples you have studied so far, can you think of rings that are not integral domains? What about C[0,1]? In Example 2, you have seen that it has zero divisors.