This is a problem from I.S.I. M.Math Subjective Sample Paper 2013 based on Maximal Ideal of a Polynomial Ring. Try this out.
Problem: Maximal Ideal of a Polynomial Ring
Let k be a field and k [x, y] denote the polynomial ring in the two variables x and y with coefficient from k . Prove that for any the ideal generated by the linear polynomials x- a and y-b is a maximal ideal of k [x, y].
Suppose I = <x-a, y-b> is not the maximal ideal. Then there exist an ideal I' such that . We show that I' = k[x,y]
Suppose I' contains I properly. There there exists an element P(x, y) (that is a polynomial in x, y with coefficients from k), which is not in I but in I'.
Say . Since k[x,y] is ring of polynomials over k which is a field (hence a unique factorization domain), hence there exists unique Q(x,y) such that where G(y) is a polynomial in y over k. As x - a is linear it will not leave any power of x in the remainder expression.
Again G(y) = R(y) (y-b) + T where T is in k. As (y-b) is linear the remainder won't have any power of y.
But T is a constant polynomial in k[x,y]. Hence . Since I' contains multiplicative identity element of field k it will absorb all polynomials from k[x,y], implying I'=k[x,y]. Proved