Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
4. Let f: RS be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-¹(J) is an ideal in R that contains Kerf. (ii) Prove that if f is surjective then f(I) is an ideal in S. (iii) Gicve an example in which if f is not surjective then f(1) need not be an ideal in S. (Hint: For (i), (ii), use the ideal test. For (iii), consider a map Z→ R, n→n and an ideal 2Z in Z.)

4. Let f: RS be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-¹(J) is an ideal in R that contains Kerf. (ii) Prove that if f is surjective then f(I) is an ideal in S. (iii) Gicve an example in which if f is not surjective then f(1) need not be an ideal in S. (Hint: For (i), (ii), use the ideal test. For (iii), consider a map Z→ R, n→n and an ideal 2Z in Z.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
plz solve with well explanation and get multiple upvotes
4. Let f : R → S be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-'(J) is an ideal in R that contains Kerf. (ii) Prove that if f is surjective then f(I) is an ideal in S. (iii) Gicve an example in which if f is not surjective then f(I) need not be an ideal in S. (Hint: For (i), (ii), use the ideal test. For (iii), consider a map Z → R, n → n and an ideal 2Z in Z.)
Transcribed Image Text:4. Let f : R → S be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-'(J) is an ideal in R that contains Kerf. (ii) Prove that if f is surjective then f(I) is an ideal in S. (iii) Gicve an example in which if f is not surjective then f(I) need not be an ideal in S. (Hint: For (i), (ii), use the ideal test. For (iii), consider a map Z → R, n → n and an ideal 2Z in Z.)
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS