Let be a ring. A subset of is said to be multiplicatively closed if , and whenever , then . In other words, is a multiplicative set where the multiplication defined on is the multiplication inherited from .
For example, let , the set is multiplicatively closed for any positive integer . Another simple example is the set , if is unital.
Remarks. Let be a commutative ring.
In the example above, if and has no divisors, then is saturated.
Assume is commutative. is saturated multiplicatively closed and iff is a union of prime ideals in .
This can be shown as follows: if let be a union of prime ideals in and . if , then for some prime ideal . Therefore, either or . This contradicts the assumption that . So is multiplicatively closed. If with , then for some prime ideal , which implies also. This contradicts the assumption that . This shows that is saturated. Of course, , since lies in any ideal of .
Conversely, assume is saturated multiplicatively closed and . For any , we want to find a prime ideal containing such that . Once we show this, then take the union of these prime ideals and that is immediate. Let be the principal ideal generated by . Since is saturated, . Let be the set of all ideals containing and disjoint from . is non-empty by construction, and we can order by inclusion. So is a poset and Zorn’s lemma applies. Take any chain in containing and let be the maximal element in . Then any ideal larger than must not be disjoint from , so is prime by the second remark in the first set of remarks. ∎
- 1 I. Kaplansky, Commutative Rings. University of Chicago Press, 1974.
|Date of creation||2013-03-22 17:29:15|
|Last modified on||2013-03-22 17:29:15|
|Last modified by||CWoo (3771)|
|Defines||saturated multiplicatively closed|