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.
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