nLab multiplicative system

In a ring

A multiplicative system in a ring RR is a set SRS\subset R which is a submonoid of the multiplicative monoid of RR, i.e. 1S1\in S and if a,bSa,b\in S then abSa b\in S.

If RR is commutative, then a multiplicative system is precisely what is necessary to construct the localization R[S 1]R[S^{-1}] as formal ‘fractions’ in RR. If RR is an integral domain, and S=R{0}S=R\setminus \{0\}, then R[S 1]R[S^{-1}] is its field of fractions.

If RR 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.

In a category

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.

category: algebra

Last revised on June 26, 2023 at 12:02:45. See the history of this page for a list of all contributions to it.