A multiplicative closed subset containing a unit element in a monoid is a left Ore set if it satisfies the
(left cancellability) If for and , then such that .
(left Ore condition) For any and there , such that .
Sometimes both conditions are called “Ore conditions”. Notice that in both conditions the new elements whose existence is equired are on the left.
If is a left Ore set in a monoid than there is a well-defined equivalence relation on pairs such that the set of equivalence classes, which are denoted by becomes a monoid together with a monoid map given by is a homomorphism of monoids; moreover this monoid map satisfies a universal property, see Ore localization. The Ore localization of monoids has been generalized to categories, see category of fractions.
If is a (unital) ring and is left Ore in a multiplicative monoid underlying , then the addition on is also well defined, commutative and associative (checking all this is rather complicated) on such that the localization map is the map of rings and satisfies the universal property for the Ore localization of rings.
A right Ore (sub)set in a monoid or is a subset such that is left Ore subset in the opposite ring .
An Ore set is a subset which is simultaneously left and right Ore subset. If where and are rings is a multiplicative subset then the satisfaction of Ore conditions in and Ore conditions in are independent in general: the reason is that in a bigger ring one has simultaneously more conditions, but also a bigger set of possible solutions for the conditions. In general it is not sufficient to check the Ore condition on generators. If are two left Ore sets, it is not true in general that the image in is left Ore; if it is then automatically is left Ore in (mutually compatible left Ore sets) and is a ring canonically isomorphic to .
The left and right Ore conditions for rings were introduced by O. Ore in 1931 in order to study the linear equations class of rings which are now called Ore domains.
(nlab note: there are many results on Ore conditions which are independent from the study of Ore localization; thus the entries should be separated)