I EXAMPLE 1 The ring of integers is an integral domain. I EXAMPLE 2 The ring of Gaussian integers Z[i] = {a + bi | a, b E Z} is an integral domain. I EXAMPLE 3 The ring Z[x] of polynomials with integer coefficients is an integral domain. I EXAMPLE 4 The ring Z[V2] = {a + bv2 1 a, b E Z} is an integral domain. I EXAMPLE 5 The ring Z, of integers modulo a prime p is an integral domain. I EXAMPLE 6 The ring Z, of integers modulo n is not an integral domain when n is not prime. I EXAMPLE 7 The ring M,(Z) of 2 × 2 matrices over the integers is not an integral domain. I EXAMPLE 8 ZOZ is not an integral domain.

I EXAMPLE 1 The ring of integers is an integral domain. I EXAMPLE 2 The ring of Gaussian integers Z[i] = {a + bi | a, b E Z} is an integral domain. I EXAMPLE 3 The ring Z[x] of polynomials with integer coefficients is an integral domain. I EXAMPLE 4 The ring Z[V2] = {a + bv2 1 a, b E Z} is an integral domain. I EXAMPLE 5 The ring Z, of integers modulo a prime p is an integral domain. I EXAMPLE 6 The ring Z, of integers modulo n is not an integral domain when n is not prime. I EXAMPLE 7 The ring M,(Z) of 2 × 2 matrices over the integers is not an integral domain. I EXAMPLE 8 ZOZ is not an integral domain.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 11E: Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R...

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