equivalent defining conditions on a Noetherian ring
Let be a ring. Then the following are equivalent:
. Let be an ascending chain of left ideals in . Let be the union of all , . Then is a left ideal, and hence finitely generated, by, say, . Now each belongs to some . Take the largest of these, say . Then for all , and therefore . But by the definition of , the equality follows.
. Let be a non-empty family of left ideals in . Since is non-empty, take any left ideal . If is maximal, then we are done. If not, must be non-empty, such that pick from this collection so that (we can find such , for otherwise would be maximal). If is not maximal, pick from such that , and so on. By assumption, this can not go on indefinitely. So for some positive integer , we have for all , and is our desired maximal element.
. Let be a left ideal in . Let be the family of all finitely generated ideals of contained in . is non-empty since is in it. By assumption has a maximal element . If , then take an element . Then is finitely generated and contained in , so an element of , contradicting the maximality of . Hence , in other words, is finitely generated. ∎
A ring satisfying any, and hence all three, of the above conditions is defined to be a left Noetherian ring. A right Noetherian ring is similarly defined.
|Title||equivalent defining conditions on a Noetherian ring|
|Date of creation||2013-03-22 18:04:27|
|Last modified on||2013-03-22 18:04:27|
|Last modified by||CWoo (3771)|