Multiplicatively Closed Set - Properties

Properties

For commutative rings, the complement of a prime ideal is an especially important example of a multiplicatively closed set. Clearly an ideal A of a commutative ring R is prime if and only if the complement R\A is multiplicatively closed. In fact, complements of prime ideals enjoy another property: that of being "saturated". A set is said to be saturated if every divisor of x in the set is also in the set (i.e., if xy is in the set, then x and y are in the set). For a commutative ring the converse is not always true: a saturated multiplicative set may not be a complement of a prime ideal. However it is true that a subset S is saturated and multiplicatively closed if and only if S is the set-theoretic complement of a non-empty set-theoretic union of prime ideals, (Kaplansky 1974, p. 2, Theorem 2).

The intersection of a family of multiplicative sets is again multiplicative, and the intersection of a family of saturated sets is saturated.

Suppose S is a multiplicatively closed subset of a commutative ring R. A standard lemma due to Krull states that there exists an ideal P of R maximal with respect to having empty intersection with S, and this ideal is a prime ideal. It follows that S is a subset of the complement R\P, which is a saturated multiplicatively closed set. Thus every multiplicatively closed set is a subset of a saturated multiplicatively closed set.