/sci/ - Science & Math

Am I missing something here? My proof:

If a is a unit, take c any element, q=0 and r=a
If a=0, take c any element, q=0 and r=0
If a is irreducible, take c=a, q=1, r =0
If a is not irreducible, unit or 0 then (since EDs are UFDs) look at its unique factorization into irreducibles and take c to be any irreducible, q the rest of the factorization and r=0.

This covers all the cases and I haven't used at all that R is an ED other than the fact that its also a UFD

Am I missing something here? My proof:

If a is a unit, take c any element, q=0 and r=a
If a=0, take c any element, q=0 and r=0
If a is irreducible, take c=a, q=1, r =0
If a is not irreducible, unit or 0 then (since EDs are UFDs) look at its unique factorization into irreducibles and take c to be any irreducible, q the rest of the factorization and r=0.

This covers all the cases and I haven't used at all that R is an ED other than the fact that its also a UFD.