multiplicative system

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.

The term ‘multiplicative system’ is also sometimes used for a set of morphisms in a category admitting a calculus of fractions.

Revised on April 23, 2009 21:14:15 by Mike Shulman (