classical ring of quotients
Let $R$ be a ring. An element of $R$ is called regular^{} if it is not a right zero divisor or a left zero divisor in $R$.
A ring $Q\supset R$ is a left classical ring of quotients for $R$ (resp. right classical ring of quotients for $R$) if it satisifies:

•
every regular element of $R$ is invertible^{} in $Q$

•
every element of $Q$ can be written in the form ${x}^{1}y$ (resp. $y{x}^{1}$) with $x,y\in R$ and $x$ regular.
If a ring $R$ has a left or right classical ring of quotients, then it is unique up to isomorphism^{}.
If $R$ is a commutative^{} integral domain^{}, then the left and right classical rings of quotients always exist – they are the field of fractions^{} of $R$.
For noncommutative rings, necessary and sufficient conditions are given by Ore’s Theorem.
Note that the goal here is to construct a ring which is not too different from $R$, but in which more elements are invertible. The first condition says which elements we want to be invertible. The second condition says that $Q$ 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 $R$ 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 
Canonical name  ClassicalRingOfQuotients 
Date of creation  20130322 14:03:01 
Last modified on  20130322 14:03:01 
Owner  mclase (549) 
Last modified by  mclase (549) 
Numerical id  5 
Author  mclase (549) 
Entry type  Definition 
Classification  msc 16U20 
Classification  msc 16S90 
Synonym  left classical ring of quotients 
Synonym  right classical ring of quotients 
Related topic  OreCondition 
Related topic  ExtensionByLocalization 
Related topic  FiniteRingHasNoProperOverrings 
Defines  regular 