Looking for some help/explanations for these problems.

1. R is a UFD => LCM's exist.

2. R is Euclidean. If a, b are associates, then phi(a)=phi(b).

3. R is a PID and S is an integral domain

phi: R--->S is onto. show either phi is an isomorphism or S is a field.

4. R is a commutative ring with a 1. Show R[x] is PID iff R is a field.

5. Prove that if R is a UFD, then R[x_1,x_2,....,X_k] (polynomials in k indeterminates) is a UFD. Exhibit a nonprincipal ideal.