nLab Ore domain

Definition

In algebra, a unital ring is an Ore domain if it is a domain $R$ in which the set $R^\times$ of all nonzero elements is an Ore set. Thus one can form the Ore localization $R[(R^\times)^{-1}]$ which is then a skew-field (division ring), called the Ore quotient ring (Ore quotient (skew)field). As Ore localizations of domains always do, it comes with a map $R\to R[(R^\times)^{-1}]$ which is 1-1. For most purposes, one sided Ore condition is sufficient, hence one considers also the weaker notions of left and right Ore domains.

Related concepts

• Dedekind domain

category: algebra