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

2 Let F be a field and let R = F[x, y] be the ring of polynomials 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 not maximal. [HINT: Find a surjective homomorphism F[x, y] → F[x].]

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

Expert Solution

Colby Messinger answered 11 months ago

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