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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..?

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    Yes, your question is addressed here. http://en.wikipedia.org/wiki/Unit_(ring_theory)2012-08-09
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    Sorry, missed the word "finite." Yes, you are right, there are no finite UFDs that are not fields, and hence they are uninteresting.2012-08-09

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