a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that S is an integral domain if and only if kerf is a prime ideal of R.
a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that S is an integral domain if and only if kerf is a prime ideal of R.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 31E: Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set...
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![Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings.
Prove that S is an integral domain if and only if kerf is a prime ideal of R.
b. Let R be a nontrivial ring such that , for each 0 + a E R there exists unique element x in R
such that axa = a. Prove that R is a division ring.
c. Check whether Z(V-5) is a Euclidean domain?
а.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0633f959-5e84-4466-98ec-7d9c4e0fb9bf%2Fad353454-ee1c-49b8-afe2-b3120a0b18e6%2Fptl2j8e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings.
Prove that S is an integral domain if and only if kerf is a prime ideal of R.
b. Let R be a nontrivial ring such that , for each 0 + a E R there exists unique element x in R
such that axa = a. Prove that R is a division ring.
c. Check whether Z(V-5) is a Euclidean domain?
а.
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