Talk:Existence of Field of Quotients

Is it proved somewhere that the operations on the field of quotients are well defined? I'm proving the same thing for Localization of Ring Exists so perhaps the proofs can be lemma'd into one page. --Linus44 20:58, 7 May 2011 (CDT)

Pretty sure it is. It's derived from the results on inverse-completions which AFAIK pretty solid. I'll take a look in a bit - only just got up, waiting for the caffeine to kick in. --prime mover 02:41, 8 May 2011 (CDT)

We have links on this page to Addition of Division Products and Product of Division Products, both of which specify the operations as well-defined on the group of units. As the field of quotients is the extension to the ring to make sure that all the elements are provided with inverses, then all the elements are units and therefore the operations as defined apply to all elements, and the well-definedness applies from there.

If we can extract the appropriate juice from that proof of well-definedness in Addition of Division Products and Product of Division Products (ouch, just noticed that Product of Division Products does not prove well-definedness) then that might be a direction to take this. --prime mover 03:41, 8 May 2011 (CDT)