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
