Rings of Residues. S. F. Ellermeyer. September 18, ; [1] m

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1 Rings of Residues S F Ellermeyer September 18, 2006 If m is a positive integer, then we obtain the partition C = f[0] m ; [1] m ; : : : ; [m 1] m g of Z into m congruence classes (This is discussed in detail in the course notes that accompany Sections 11 and 21) Our goal is to de ne binary operations of addition and multiplication on the members of C De nition 1 A binary operation on a set, A, is a function mapping AA into A For example, the normal addition of integers is a binary operation on the set of integers because when we add two integers, we obtain another integer For example = 63 We can think of addition of integers as being a function called + mapping ZZ into Z where, for example, + (7; 56) = 63 Clearly, normal multiplication is also a binary operation on the integers Our operations of addition and multiplication on members of C will be de ned as follows: De nition 2 For any two members, [a] m and [b] m, of C, we de ne [a] m + [b] m = [a + b] m (where the + on the right hand side of this equation is the normal operation of addition of integers) and [a] m [b] m = [a b] m (where the on the right hand side of this equation is the normal operation of multiplication of integers) 1

2 Example 3 Compute [3] 6 + and [3] 6 Solution 4 [3] 6 + = [3 + 5] 6 = [8] 6 and [3] 6 = [3 5] 6 = [15] 6 Since [a] m and [b] m are sets, the operations de ned in De nition 2 make sense only if the results of the addition and multiplication do not depend on the particular choice of representative from [a] m and [b] m In other words, we need to assure ourselves that the operations of addition and multiplication that we have de ned are well de ned Some things that we have learned previously (in Section 21) allow us to conclude that these operations are indeed well de ned In particular, we know that x y mod m if and only if [x] m = [y] m and z w mod m if and only if [z] m = [w] m We also know that if a c mod m and b d mod m, then and (a + b) (c + d) mod m a b c d mod m We want to make sure that if [a] m = [c] m and [b] m = [d] m, then [a] m + [b] m = [c] m + [d] m Taking the above considerations into account, we see that this is in fact true because [a] m + [b] m = [a + b] m (by de nition) = [c + d] m (because (a + b) (c + d) mod m) = [c] m + [d] m (by de nition) By similar reasoning, we can also rest assured that multiplication, as de ned in De nition 2, is well de ned Example 5 In Example 3, we computed [3] 6 + = [3 + 5] 6 = [8] 6 and [3] 6 = [3 5] 6 = [15] 6 We will show that the same results are obtained when di erent representatives of [3] 6 and [4] 6 are used in doing these computations Note that 33 3 mod 6 and 17 5 mod 6 Thus [33] 6 = [3] 6 and [17] 6 = When we 33 and 17 as the representatives in our computations, we obtain [33] 6 + [17] 6 = [50] 6 However, both [8] 6 and [50] 6 are equal to [2] 6 Thus [33] 6 + [17] 6 = [3] 6 + 2

3 For the multiplication, we obtain [33] 6 [17] 6 = [561] 6 Since [15] 6 and [561] 6 are both equal to [3] 6, we see that [33] 6 [17] 6 = [3] 6 Recalling that every congruence class modulo m has a principal representative, a mod m (where 0 a mod m < m), let us agree to simplify things by always using the principal representative of any congruence class, [a] m, to represent [a] m Having agreed to this, we can simplify notation by omitting the brackets and the subscript m in doing addition and multiplication on elements of C (as long as the value of m is understood) For example, as long as it is understood that we are operating modulo 6, instead of writing [3] 6 + = [2] 6, we can simply write = 2, and instead of writing [3] 6 = [3] 6, we can just write 3 5 = 3 De nition 6 For a given positive integer m, the ring of residues modulo m is de ned to be the set A = f0; 1; : : : ; m 1g with binary operations of addition and multiplication as de ned above (That is, addition and multiplication are performed modulo m) The ring of residues modulo m is denoted by Z m Remark 7 The ring of residues modulo m is not a set of numbers! It actually consists of three things: 1 the set of numbers A = f0; 1; : : : ; m 1g 2 a binary operation called addition and denoted by + 3 a binary operation called multiplication and denoted by Thus Z m = fa; +; g Nonetheless, we will often abuse terminology and refer to Z m as though it is the set A For example, we will say things like Let x be a member of Z m and write x 2 Z m when what we really mean to say is that x is a member of the set A which is the underlying set of numbers that comprises Z m (but this is too long to say) 3

4 Example 8 Let us write down the addition and multiplication tables for Z 6 A Useful Diagram of the Ring Z As can be seen from the tables constructed in Example 8, there are some algebraic properties that hold in Z that do not hold in Z 6 For example, in Z, if we have ab = 0, then it must be true that either a = 0 or b = 0 This is not true in Z 6 though because, for example, 3 2 = 0 Also, in Z, if a b = a 4

5 and a 6= 0, then it must be true that b = 1 However this is not true in Z 6 because, for example, 3 3 = 3 Another interesting feature that contrasts Z and Z 6 is that the only numbers in Z that have multiplicative inverses in Z are 1 and 1; whereas there is a number not equal to 1 in Z 6 that has a multiplicative inverse In particular, since 5 5 = 1 in Z 6, we see that 5 has a multiplicative inverse (which is itself) Example 9 Let us write down the addition and multiplication tables for Z There are some important di erences between Z 5 and Z 6 that should be pointed out: 1 Notice that the only way to have a b = 0 in Z 5 is if a = 0 or b = 0 This is not true in Z 6 2 Notice that every non zero member of Z 5 has a multiplicative inverse For example, in Z 5, the multiplicative inverse of 2 is 3 because 23 = 1 However, not every non zero member of Z 6 has a multiplicative inverse (Only 5 has has a multiplicative inverse in Z 6 ) The following proposition lists some of the basic properties that are common to all rings of residues Proposition 10 In any ring of residues (modulo m), the following properties hold for any elements a, b, and c: 1 (associative properties of addition and multiplication) a + (b + c) = (a + b) + c and a (b c) = (a b) c 2 (commutative properties of addition and multiplication) a + b = b + a and a b = b a 5

6 3 (existence of an additive and multiplicative identity elements) a+0 = a and a 1 = a 4 (existence of additive inverses) For each element a, there exists an element a, called the additive inverse of a, such that a + ( a) = 0 5 (distributive property) a (b + c) = a b + a c Proof We will prove parts 1 and 5 of the proposition and leave the proofs of the other parts as homework 1 Let a, b, and c be elements of Z m Then a + (b + c) = [a] m + [b + c] m (converting from simpli ed notation) = [a + (b + c)] m (by de nition of addition in Z m ) = [(a + b) + c] m (by the associative property of addition for Z) = [a + b] m + [c] m (by de nition of addition in Z m ) = (a + b) + c (converting back to simpli ed notation) 5 Let a, b, and c be elements of Z m Then a (b + c) = [a] m [b + c] m = [a (b + c)] m = [a b + a c] m = [a b] m + [a c] m = a b + a c We will now de ne two terms that are obviously relevant to our study of the rings Z m De nition 11 An element, a, of Z m is called a zero divisor if a 6= 0 and there exists an element, x, of Z m such that a x = 0 De nition 12 An element, a, of Z m is called a unit if a has a multiplicative inverse in Z m In other words, a is a unit if there exists an element x in Z m such that a x = 1 6

7 Example 13 By referring to the multiplication tables in Examples 8 and 9, we see that the zero divisors in Z 6 are 2, 3, and 4 and that the units in Z 6 are 1 and 5 In Z 5, there are no zero divisors and every non zero member of Z 5 is a unit We now discuss how to determine if a given element in Z m is a zero divisor or a unit To determine if a given non zero element a 2 Z m is a zero divisor, we must determine whether or not there any non zero solutions to the equation a x = 0 Since this equation can be written as [ax] m = [0] m, we observe that we are looking for non zero solutions of the congruence equation ax 0 mod m The solution set of this equation is [0] m= gcd(a;m) If gcd (a; m) = 1, then the solution set is simply [0] m which means that x = 0 is the only solution of a x = 0 and hence that a is not a zero divisor On the other hand, if gcd (a; m) > 1, then [0] m= gcd(a;m) is the union of gcd (a; m) congruence classes modulo m All but one of the congruence classes in this union does not contain 0 Therefore, in this case, there are non zero solutions to the equation a x = 0, and this means that a is a zero divisor In summary, a is a zero divisor if and only if gcd (a; m) > 1 To determine if a given non zero element a 2 Z m is a unit, we must determine whether or not there any non zero solutions to the equation ax = 1 Since this equation can be written as [ax] m = [1] m, we observe that we are looking for solutions of the congruence equation ax 1 mod m If gcd (a; m) = 1, then this equation does have solutions (because, in this case, gcd (a; m) divides 1) and hence a x = 1 has a solution, meaning that a is a unit On the other hand, if gcd (a; m) > 1, then gcd (a; m) does not divide 1 and the congruence equation ax 1 mod m does not have any solutions This means that the equation a x = 1 does not have a solution and hence a is not a unit In summary, a is a unit if and only if gcd (a; m) = 1 We now see how to tell whether or not a given member of Z m is a unit or a zero divisor 1 If a 6= 0 and gcd (a; m) = 1, then a is a unit and not a zero divisor 2 If a 6= 0 and gcd (a; m) > 1, then a is a zero divisor and not a unit Note that any non zero element of Z m must be either a zero divisor or a unit and cannot be both Also note that if p is a prime number, then gcd (a; p) = 1 for all a such that 1 a < p This means that every non zero 7

8 element of Z p is a unit and that Z p has no zero divisors (We have already seen this illustrated in our study of Z 5 ) Example 14 In Z 8, since gcd (1; 8) = 1 gcd (2; 8) = 2 gcd (3; 8) = 1 gcd (4; 8) = 4 gcd (5; 8) = 1 gcd (6; 8) = 2 gcd (7; 8) = 1, we see that the zero divisors in Z 8 are 2; 4; and 6 and that the units in Z 8 are 1; 3; 5; and 7 Note, for example, that in Z 8 we have 2 4 = 0 and 3 3 = 1 (so 3 is its own multiplicative inverse in Z 8 ) Exercise 15 Make addition and multiplication tables for Z 1, Z 2, Z 3, Z 4, Z 7, and Z 8 Identify the zero divisors and the units of each of these rings Exercise 16 In the textbook, Section 24 (pages 80 and 81), do problems 2, 3, 5, 6, 11, 12, and 14 8

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