# Answer to Question #15377 in Abstract Algebra for Ross

Question #15377

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]

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]

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