Localization at a submonoid
Let be a commutative unital ring and be a submonoid of not containing zero (in other words and is closed under multiplication). Such a is also termed a multiplicatively closed subset. The localization of at , denoted or , is defined as follows:
- Set-theoretically, it is the set of ordered pairs with and modulo the following equivalence relation:
The pair is denoted as \r . is termed the numerator and the denominator of the fraction .
- The operations are as follows:
(It needs to be checked that these operations go down under the equivalence).
is a commutative unital ring and is the subring comprising those elements with denominator .
- Localization at a prime ideal: The terminology is somewhat misleading, because this actually means localization with respect to the submonoid which is the set-theoretic complement of the prime ideal.
- Localization at a maximal ideal: The same as localization at a prime ideal, only for the case where the prime ideal is in fact a maximal ideal.
- Field of fractions: When the ring is an integral domain, we can choose the submonoid as the set of all nonzero elements (equivalently, we are localizing at the zero ideal). The ring obtained is a field, termed the field of fractions of the integral domain.
- Total quotient ring: Here, the submonoid is the set of all nonzerovisors. This generalizes the notion of field of fractions to rings that are not integral domains.