Let be a set of representatives for each irreducible element.

For be non-zero and , define so that but .

Then where is unit.

Let be non-zero so .

Where the product is over .

Notice that where is a LCM.

What is ?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.

It should be a known result that being a field implies that is a PID.4. R is a commutative ring with a 1. Show R[x] is PID iff R is a field.

Now if is a PID, let be in .

For the other direction see this.

If is an UFD then is an UFD[/tex].5. Prove that if R is a UFD, then R[x_1,x_2,....,X_k] (polynomials in k indeterminates) is a UFD.

Once you know this you can induct.

Consider . And consider the set of all polynomials that have constant term equal to zero. This set is an ideal but it is a non-pricipal ideal.Exhibit a nonprincipal ideal