Abstract Algebra Theory and Applications - Computer Science ...

234 CHAPTER 14 RINGSExample 3. We can define the product of two elements a and b in Z n by ab(mod n). For instance, in Z 12 , 5 · 7 ≡ 11 (mod 12). This product makes theabelian group Z n into a ring. Certainly Z n is a commutative ring; however,it may fail to be an integral domain. If we consider 3 · 4 ≡ 0 (mod 12) inZ 12 , it is easy to see that a product of two nonzero elements in the ring canbe equal to zero.A nonzero element a in a ring R is called a zero divisor if there is anonzero element b in R such that ab = 0. In the previous example, 3 and 4are zero divisors in Z 12 .Example 4. In calculus the continuous real-valued functions on an interval[a, b] form a commutative ring. We add or multiply two functions by addingor multiplying the values of the functions. If f(x) = x 2 and g(x) = cos x,then (f +g)(x) = f(x)+g(x) = x 2 +cos x and (fg)(x) = f(x)g(x) = x 2 cos x.Example 5. The 2 × 2 matrices with entries in R form a ring underthe usual operations of matrix addition and multiplication. This ring isnoncommutative, since it is usually the case that AB ≠ BA. Also, noticethat we can have AB = 0 when neither A nor B is zero.Example 6. For an example of a noncommutative division ring, let( ) ( )1 00 11 =i =0 1−1 0j =( 0 ii 0)k =( i 00 −i),where i 2 = −1. These elements satisfy the following relations:i 2 = j 2 = k 2 = −1ij = kjk = iki = jji = −kkj = −iik = −j.Let H consist of elements of the form a + bi + cj + dk, where a, b, c, d arereal numbers. Equivalently, H can be considered to be the set of all 2 × 2