Logo
Home Study Material Contact Search
AI Chat FREE
  1. mathematics
  2. advance math
  3. 2 let f be a field and let r = f[x, y] be the ring

2 Let F be a field and let R = F[x, y] be the ring...

60.1K

Verified Solution

Question

Advance Math

2 Let F be a field and let R = F[x, y] be the ring ofpolynomials in two variables with coefficients in F.

(a) Prove that

ev(0,0) : F[x, y] ? F

p(x, y) ? p(0, 0)

is a surjective ring homomorphism.

(b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x,y) + ys(x, y) | r,s ? F[x, y]}

(c) Use the first isomorphism theorem to prove that (x, y) ?F[x, y] is a maximal ideal.

(d) Find an ideal I ? F[x, y] such that I is prime but notmaximal. [HINT: Find a surjective homomorphism F[x, y] ? F[x].]

(e) Find an ideal J ? F[x, y] such that J is not prime

Answer & Explanation Solved by verified expert
3.8 Ratings (557 Votes)
    See Answer
Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:
  • Unlimited Question Access with detailed Answers
  • Zin AI - 3 Million Words
  • 10 Dall-E 3 Images
  • 20 Plot Generations
  • Conversation with Dialogue Memory
  • No Ads, Ever!
  • Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students

Biology
1.2K views

Think of the current issue that you want to present in your multimodal advocacy campaign...

Medical Sciences
1.2K views

What are neurotransmitters; how are neurotransmitters classified and what are the classes (i.e. chemical classes versus...

Biology
1.2K views

how to do a research project on oil, gas & solidsh?

Psychology
1.2K views

Resume

Accounting
1.2K views

Questions Jones Company is a merchandise company that uses the perpetual inventory method.  The business prepares financial statements on...

Algebra
1.2K views

Taylor is running a 26.2-mile marathon. Let x represent the number of miles Taylor has...

Accounting
1.2K views

Wriston Company has $220,000 to invest. The...

Accounting
1.2K views

Alex and Bess have been in partnership for many years. The partners, who share profits...

Accounting
1.2K views

Prepare the following adjusting entries at August 31 for Walgreens. (a) Interest on notes payable...

Accounting
1.2K views

47. Alan and Sara Winthrop are a married couple who file a joint income tax...

Accounting
1.2K views

Assess whether Samsung could benefit by using a standard cost system. Your evaluation should include...

Accounting
1.2K views

Which of the following is for the specific purpose of reporting cash flows? ...