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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fa992e3c4-2fad-4455-96b6-10e5dadc9be4%2Fkgj6ybh_processed.png&w=3840&q=75)
Transcribed Image Text: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.
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