When studying UFDs I started to get confused... If $u$,$v$ are units in $R$ then $u^{-1}$$v$ is a unit in $R$ and so $v$ = ($u$$u^{-1}$)$v$ = $u$($u^{-1}$$v$) hence u and v are associates..? Are really all units associates? So in every field all non-zero elements are associates?
What about the units in a UFD, may they be factored into irreducibles?
And also, since every UFD is an integral domain and every finite integral domain is a field, we can't really have any interesting finite UFDs? Since these are all fields..?