Polynomial Ring, Show I is prime but not maximal

[Rederick]

Homework Statement

Let R = Z[x] be a polynomial ring where Z is the integers. Let I = (x) be a principal ideal of R generated by x. Prove I is a prime ideal of R but not a maximal ideal of R.

Homework Equations

The Attempt at a Solution

I want to show that R/I is an integral domain which implies I is a prime ideal and that R/I is NOT a field which implies I is not a maximal ideal.
I'm not sure how to represent R/I to show those two things. I know R/I = f(x) +I but I don't know where to go from there.

 

[Hurkyl]

I'm not sure how to represent R/I to show those two things. I know R/I = f(x) +I but I don't know where to go from there.

For any particular polynomial f, the element represented by f(x) + I also has many other representations as g(x) + I for other polynomials g. It is often helpful to simplify things; maybe it will suggest something.

An alternative would be to guess a ring that is isomorphic to the quotient R/I, and try to write down the isomorphism, then try to prove both directions are well-defined.

If all else fails, you could try doing arithmetic in R/I to gain familiarity with it, or maybe try to prove directly from the definitions of everything involved that R/I is an integral domain that is not a field.