Let R be a commutative ring with multiplicative identity 1 0. Call an ideal I in R prime if for any x, y Є R with xy Є I, we must have either x = I or y Є I. Prove that if I is not prime, then there exists an ideal J of R with I ÇJ Ç R. [Hint: Saying I is not prime means there exist x, y Є R such that xy Є I but x, y ‡ I. Now set J=I+xR and prove that I ‡ J and J ‡ R.]
Let R be a commutative ring with multiplicative identity 1 0. Call an ideal I in R prime if for any x, y Є R with xy Є I, we must have either x = I or y Є I. Prove that if I is not prime, then there exists an ideal J of R with I ÇJ Ç R. [Hint: Saying I is not prime means there exist x, y Є R such that xy Є I but x, y ‡ I. Now set J=I+xR and prove that I ‡ J and J ‡ 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...
Related questions
Question
![Let R be a commutative ring with multiplicative identity 1 0.
Call an ideal I in R prime if for any x, y Є R with xy Є I, we must have either x = I or
y Є I. Prove that if I is not prime, then there exists an ideal J of R with I ÇJ Ç R.
[Hint: Saying I is not prime means there exist x, y Є R such that xy Є I but x, y ‡ I. Now
set J=I+xR and prove that I ‡ J and J ‡ R.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2bb1218-57c3-496c-854f-870cc28d6c36%2Fe082bb64-a6fc-4e69-ab3e-14f25cf2ccf0%2Fdcyp0qk_processed.png&w=3840&q=75)
Transcribed Image Text:Let R be a commutative ring with multiplicative identity 1 0.
Call an ideal I in R prime if for any x, y Є R with xy Є I, we must have either x = I or
y Є I. Prove that if I is not prime, then there exists an ideal J of R with I ÇJ Ç R.
[Hint: Saying I is not prime means there exist x, y Є R such that xy Є I but x, y ‡ I. Now
set J=I+xR and prove that I ‡ J and J ‡ R.]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,