Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only if R is a field.
Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only if R is a field.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 18E: 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is...
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