classical ring of quotients
A ring is a left classical ring of quotients for (resp. right classical ring of quotients for ) if it satisifies:
every regular element of is invertible in
every element of can be written in the form (resp. ) with and regular.
If a ring has a left or right classical ring of quotients, then it is unique up to isomorphism.
Note that the goal here is to construct a ring which is not too different from , but in which more elements are invertible. The first condition says which elements we want to be invertible. The second condition says that should just enough extra elements to make the regular elements invertible.
Such rings are called classical rings of quotients, because there are other rings of quotients. These all attempt to enlarge somehow to make more elements invertible (or sometimes to make ideals invertible).
Finally, note that a ring of quotients is not the same as a quotient ring.
|Title||classical ring of quotients|
|Date of creation||2013-03-22 14:03:01|
|Last modified on||2013-03-22 14:03:01|
|Last modified by||mclase (549)|
|Synonym||left classical ring of quotients|
|Synonym||right classical ring of quotients|