nLab multiplicative system

In a ring

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

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

If $R$ 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