nLab Ore set




A multiplicative closed subset SRS\subset R containing a unit element in a monoid RR is a left Ore set if it satisfies the

  • (left cancellability) If ns=msn s = m s for n,mRn,m\in R and sSs\in S, then sS\exists s'\in S such that sn=sms' n = s' m.

  • (left Ore condition) For any rRr\in R and sSs\in S there rR\exists r'\in R, sS\exists s'\in S such that sr=rss'r = r's.

Sometimes both conditions are called “Ore conditions”. Notice that in both conditions the new elements whose existence is required are on the left.

If SS is a left Ore set in a monoid than there is a well-defined equivalence relation \sim on pairs (s,r)S×R(s,r)\in S\times R such that the set of equivalence classes, which are denoted by s 1r:=[(s,r)]S 1R:=S×R/s^{-1}r := [(s,r)]\in S^{-1}R:=S\times R/\sim becomes a monoid together with a monoid map j S:RS 1Rj_S: R\to S^{-1}R given by r1 1rr\mapsto 1^{-1}r 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 RR is a (unital) ring and SRS\subset R is left Ore in a multiplicative monoid underlying RR, then the addition on S 1RS^{-1}R is also well defined, commutative and associative (checking all this is rather complicated) on S 1RS^{-1}R 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 RR is a subset SRS\subset R such that SS is left Ore subset in the opposite ring R opR^{op}.

An Ore set is a subset SRS\subset R which is simultaneously left and right Ore subset. If SRRS\subset R\subset R' where RR and RR' are rings is a multiplicative subset then the satisfaction of Ore conditions in RR and Ore conditions in RR' 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 S,TRS,T\subset R are two left Ore sets, it is not true in general that the image i T(S)i_T(S) in T 1RT^{-1}R is left Ore; if it is then automatically i S(T)i_S(T) is left Ore in S 1RS^{-1}R (mutually compatible left Ore sets) and (i S(T)) 1S 1R(i_S(T))^{-1}S^{-1} R is a ring canonically isomorphic to (i T(S)) 1T 1T(i_T(S))^{-1}T^{-1}T.

The left and right Ore conditions for rings were introduced by Øystein 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)


Last revised on February 9, 2023 at 06:19:23. See the history of this page for a list of all contributions to it.