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?
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