Let R = C[x, y] be the ring of polynomials in two variables over C. Let P = (x) be the ideal of R generated by x. Define a map 0: R → C[y] by 0(p(x, y)) = p(0, y). In other words, given a polynomial in two variables x and y, we plug in 0 for x to get a polynomial in y. (a) Isa ring homomorphism? What is the kernel and the image of 0? (b) Is P a prime ideal? Is P a maximal ideal? (c) Is C[x, y] a PID?

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