mul uhity and no zer Thus, in an integral domain, a product is 0 only when one of the factors is 0; that is, ab = 0 only when a = 0 or b : examples show that many familiar rings are integral domains...

Which of examples 1-5 are fields?

Extracted text: mul uhity and no zer Thus, in an integral domain, a product is 0 only when one of the factors is 0; that is, ab = 0 only when a = 0 or b : examples show that many familiar rings are integral domains and some familiar rings are not. For each example, the student should verify the 0. The following assertion made. I EXAMPLE 1 The ring of integers is an integral domain. Hoiry 255


Extracted text: mead 256 Rings EXAMPLE 2 The ring of Gaussian integers Z[i] = {a + bi \a, bEZ is an integral domain. EXAMPLE 3 The ring Z[x] of polynomials with integer coefficients is an integral domain. EXAMPLE 4 The ring Z[V2] = {a + bV2l a, b E Z} is an integral domain. EXAMPLE 5 The ring Z, of integers modulo a prime p is an integral р domain. EXAMPLE 6 The ring Z, of integers modulon is not an integral do- main when n is not prime. п Th EXAMPLE 7 The ring M,(Z) of 2 x 2 matrices over the integers is not an integral domain. EXAMPLE 8 ZOZ is not an integral domain. What makes integral domains particularly appealing is that they have an important multiplicative group theoretic property, in spite of the la that the nonzero elements need not form a group under multiplication. This property is cancellation. Theorem 13.1 Cancellation IC Let a, b, and c belong to an integral domain. If a # 0 and ab = ac, then b c. PROOF From ah