Let be a ring. A subset of is said to be an -system if
for every , there is an , such that .
Form the set . In addition, inductively define
and form . Then both and are -systems (as well as -systems). Furthermore, is an -system which is not an -system.
The example above suggests that, given an -system and any , we can “construct” an -system such that . Start with , inductively define , where the existence of such that is guaranteed by the fact that is an -system. Then the collection is a subset of that is an -system. For if we pick any and , if , then is both the left and right sections of , meaning that there are such that (this can be easily proved inductively). As a result, , and .
Suppose is semiprime. Let . Then , which means there is an element such that . So is an -system. Now suppose that is an -system. Let with the condition that . This means for all . If , then there is some with , contradicting condition on . Therefore, , and is semiprime. ∎
|Date of creation||2013-03-22 17:29:29|
|Last modified on||2013-03-22 17:29:29|
|Last modified by||CWoo (3771)|