Answer to Question #15377 in Abstract Algebra for Ross

Question

Let Z[x] be the domain of all polynomials with integer coefficients. Consider the constant polynomial 2 and the polynomial x. Let S = {a(x)*2 + b(x)*x: a(x),b(x) w/in Z[x]}. (note: Z is integers symbol)
a)Prove S = {f(x) w/in Z[x]: f(0) w/in 2Z}
b)Prove S is an ideal in Z[x]
c)Prove there is no polynomial d(x) w/in Z[x] such thats S = {q(x)d(x): q(x) w/in Z[x]}. [This Z[x] is not a principle ideal domain]