MATH Algebra for High School Teachers Units and Zero Divisors

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1 MATH Algebra for High School Teachers Units and Zero Divisors Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 9

2 Examples We saw previously that every element of Z m has an additive inverse in Z m, but not every (non-zero) element necessarily has a multiplicative inverse. How does this compare to more familiar number systems, such as the integers (Z) or rational numbers (Q)? Both Z and Q have all of the properties we stated for Z m, including an additive identity (0), additive inverses ( a), and a multiplicative identity (1). If a and b are integers such that ab = 1, then a = b = ±1. Hence 1 and 1 are the only elements of Z that have a multiplicative inverse in Z. If a b is a non-zero rational number, so a and b are non-zero integers, then b a is in Q and a b b a = 1. Hence every non-zero element of Q has a multiplicative inverse in Q. A very important property of the integers and rational numbers is that if a and b are elements of Z or Q and ab = 0, then a = 0 or b = 0. D.L. White (Kent State University) 2 / 9

3 Examples Z Z Recall our observations about Z 4 and Z 5 : Not every row of the multiplication table of Z 4 contains a 1. Conclusion: Not every element of Z 4 has a multiplicative inverse. Every non-zero row of the multiplication table for Z 5 does contain a 1. Conclusion: Every non-zero element of Z 5 has a multiplicative inverse. There is a 0 in the table for Z 4, in neither the 0 row nor the 0 column. Conclusion: Products of non-zero elements can be 0. We will now determine which elements of Z m have multiplicative inverses and which non-zero elements of Z m can be multiplied together to get 0. D.L. White (Kent State University) 3 / 9

4 Units and Inverses Definition Let [a] be an element of Z m. If there is an element [b] in Z m such that [a][b] = [1], we say that [a] is a unit in Z m and [b] is a multiplicative inverse of [a]. In this case, we write [b] = [a] 1. In Z 4, we have [1] 1 = [1], [3] 1 = [3], but [2] does not have a multiplicative inverse. In Z 5, we have [1] 1 = [1], [2] 1 = [3], [3] 1 = [2], [4] 1 = [4]. D.L. White (Kent State University) 4 / 9

5 Units and Inverses Note that in Z m, [a][b] = [1] [ab] = [1] ab 1 (mod m). Hence [b] is a multiplicative inverse for [a] in Z m if and only if b is a multiplicative inverse for a mod m. Recall that a has a multiplicative inverse mod m if and only if (a, m) = 1. Therefore, we have the following theorem. Theorem An element [a] of Z m is a unit if and only if (a, m) = 1. Definition The set U m = {[a] Z m (a, m) = 1} of all units in Z m is called the group of units of Z m. Note that if (a, m) = 1 and (b, m) = 1, then (ab, m) = 1 [exercise]. Hence if [a] and [b] are units in Z m, then [a][b] = [ab] is a unit in Z m ; that is, the set U m is closed under multiplication. Exercise: Express ([a][b]) 1 in terms of [a] 1 and [b] 1. D.L. White (Kent State University) 5 / 9

6 Example U 15 = {[a] Z 15 (a, 15) = 1} = {[1], [2], [4], [7], [8], [11], [13], [14]} Verify that in Z 15, [1] 1 = [1], [2] 1 = [8], [8] 1 = [2], [4] 1 = [4], [7] 1 = [13], [13] 1 = [7], [11] 1 = [11], [14] 1 = [14]. Check that U 15 is closed under multiplication; for example, [2][11] = [22] = [7] [7][14] = [98] = [8] [4][8] = [32] = [2], etc. Observe that U 15 is not closed under addition; for example, [2] + [4] = [6] is not in U 15. D.L. White (Kent State University) 6 / 9

7 Zero Divisors An important difference between Z and Z m is the possibility in Z m that there are non-zero elements whose product is 0, as we saw in Z 4. Definition An element [a] of Z m is a zero divisor if [a] [0] and there is an element [b] [0] in Z m such that [a][b] = [0]. NOTE: By definition, [0] is not a zero divisor. Example: In Z 10, [2][5] = [10] = [0] [4][5] = [20] = [0] [6][5] = [30] = [0] [8][5] = [40] = [0], and so [2], [4], [5], [6], and [8] are zero divisors in Z 10. Observe that [1], [3], [7], and [9] are units in Z 10 and are not zero divisors. D.L. White (Kent State University) 7 / 9

8 Zero Divisors In fact, a unit can never be a zero divisor. If [a] is a unit in Z m and [a][b] = [0] for some [b] in Z m, then [a][b] = [0] [a] 1 ([a][b]) = [a] 1 [0] ([a] 1 [a])[b] = [0] [1][b] = [0] [b] = [0]. Hence there is no non-zero [b] in Z m such that [a][b] = [0], and so [a] is not a zero divisor. D.L. White (Kent State University) 8 / 9

9 Zero Divisors It follows that if [a] is a zero divisor in Z m, then (a, m) 1. Conversely, if (a, m) 1, then 1 m [ ] < m. m Hence m, and so m [0]. But we have [ ] [ ] [ ] [ ] [a] m = am = a [m] = a [0] = [0]. Thus if m a, so [a] [0], and (a, m) 1, then [a] is a zero divisor in Z m. We therefore have the following theorem. Theorem An element [a] of Z m is a zero divisor if and only if m a and (a, m) 1. Hence there are three disjoint subsets of Z m : {[0]}: all [a] such that m a. Units: all [a] such that (a, m) = 1. Zero divisors: all [a] such that m a and (a, m) 1. D.L. White (Kent State University) 9 / 9