A multiplicative system in a ring is a set which is a submonoid of the multiplicative monoid of , i.e. and if then .
If is commutative, then a multiplicative system is precisely what is necessary to construct the localization as formal ‘fractions’ in . If is an integral domain, and , then is its field of fractions.
If is not commutative, then one generally needs extra conditions, such as the Ore condition, in order to construct a localization. Still, if the Ore condition is required, we usually state this and call it a 2-sided Ore set, a denominator set or alike.
The term ‘multiplicative system’ is often sometimes used for a set of morphisms in a category admitting a calculus of fractions hence it satisfies the Ore conditions.
Last revised on June 26, 2023 at 12:02:45. See the history of this page for a list of all contributions to it.