You are here
Homeclassical ring of quotients
Primary tabs
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. $yx^{{1}}$) 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 contain 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.
Mathematics Subject Classification
16U20 no label found16S90 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Jun 13
new question: young tableau and young projectors by zmth
Jun 11
new question: binomial coefficients: is this a known relation? by pfb
Jun 6
new question: difference of a function and a finite sum by pfb