Let be a ring. A subset of is called an -system if
for every two elements , there is an element such that .
-Systems are a generalization of multiplicatively closet subsets in a ring. Indeed, every multiplicatively closed subset of is an -system: any , then , hence . However, the converse is not true. For example, the set
is an -system, but not multiplicatively closed in general (unless, for example, if ).
Remarks. -Systems and prime ideals of a ring are intimately related. Two basic relationships between the two notions are
An ideal in a ring is a prime ideal iff is an -system.
is prime iff implies or , iff implies that there is with iff is an -system. ∎
Given an -system of and an ideal with . Then there exists a prime ideal with the property that contains and , and is the largest among all ideals with this property.
Let be the collection of all ideals containing and disjoint from . First, . Second, any chain of ideals in , its union is also in . So Zorn’s lemma applies. Let be a maximal element in . We want to show that is prime. Suppose otherwise. In other words, with . Then and both have non-empty intersections with . Let
where and . Then there is such that . But this implies that
as well, contradicting . Therefore, is prime. ∎
-Systems are also used to define the non-commutative version of the radical of an ideal of a ring.
|Date of creation||2013-03-22 17:29:09|
|Last modified on||2013-03-22 17:29:09|
|Last modified by||CWoo (3771)|