Consider the two homomorphisms phi_0:Z[x] −→ Z defined by 0(f(x)) = f(0) and Rho:Z −→ Z5 defined by (n) = ¯n.

We have proven that each of these maps is a ring homomorphism, and also that the composition of two ring homomorphisms is again a ring homomorphism. Therefore Psi = Rho ◦ Phi_0 is a ring homomorphism from Z[x] to Z5.

Let I = ker( Psi).

(a) Prove that I is a maximal ideal of Z[x].

Hint. Sometimes the indirect approach is easier.

(b) Recall that an ideal of a ring R is principal when it is of the form aR = <a> for some a ∈ R.

Prove that I is not a principal ideal.

Hint. Suppose that I is principal. What are the possible generators for I?

Now show that they all generate something other than I.

(c) Give a non-zero prime ideal of Z[x] that is not maximal.

Hint. What about an ideal contained in I?

**Subject Mathematics Abstract Algebra**