Homeomorphism of Rings: Proving Existence for Prime Numbers p and q

[TimNguyen]

Let p,q be two prime numbers. Prove that there exists a homeomorphism of rings such that f([1]_p)=[1]_q from Z_p[X] into Z_q[X] if and only if p=q.

I believe that the converse of the statement is trivial but the implication seems to be obvious? I really don't know what there really is to prove in this.

 

[AKG]

I think you mean homomorphism? Certainly, the converse is trivial but if the implication is obvious, then what is the proof? Note, that the statement that there exists a homomoprhism of rings such that f(1) = 1 is the statement that there exists a homomorphism of rings such that f(1) = 1.