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

Q

DEPRECIATION METHODS Charlene is evaluating a capital budgeting project that should last for 4 years. The project...

Finance
Q

1) Which of the following is a condition that represents inflammation of a herniation through the...

Medical Sciences
Q

For staffing purposes, a retail store manager would like to standardize the number of checkout lanes...

Basic Math
Q

4 Choose the correct option a No of field Lines Leaving the sphere No of...

Physics
Q

Find lim h 0 f 7 h f 7 h f 7 h f 7...

Calculus
Q

An arrow is shot straight up into the air The function H t 16t 176t...

Algebra
Q

1 O A child is about how Choose the correct O 95 mm

Algebra
Q

Using the chart answer: 1.) Compute cost of direct materials used. 2.) Compute cost of...

Accounting
Q

The historical cost concept reflects the fact that financial accounting practice favors: Reliability...

Accounting
Q

The following describes production activities of Mercer Manufacturing for the year. Actual direct materials used...

Accounting
Q

A random sample of 40 people who owned their own business reported a yearly mean...

Accounting
Q

Describe, in detail, the Dupont equation. Include in your answer each of the following: (a)...

Finance