A multiplicative subset of a ring contains and is closed under multiplication. It may contain zero, in which case its localization is the zero ring.
A multiplicative subset is saturated if every factor of a product in the subset also lies in it. We include in a multiplicative subset. A saturated subset containing is the whole ring. The units form a saturated subset in a commutative ring.
In , the saturated multiplicative subset has complement equal to the union of over all positive primes. No finite subcollection of prime ideals covers this complement: a prime outside the finite list is a nonunit not in any listed ideal.
The complement of a multiplicative subset is a union of prime ideals exactly when the subset is saturated. If , saturation ensures ; extend this to a prime ideal avoiding a multiplicative subset. Conversely, a product in cannot have a factor in a prime ideal contained in the complement.
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In the context of mathematics, particularly in the fields of algebra and number theory, a **multiplicatively closed set** is a subset of a given set that is closed under the operation of multiplication. This means that if you take any two elements from this set and multiply them together, the result will also be an element of the set. Formally, let \( S \) be a set.