properties of nil and nilpotent ideals
Let be ideals of a ring . If is nil and is nil, then is nil. If is nilpotent and is nilpotent, then is nilpotent.
Suppose that and are nil. Let . Then for some , since is nil. But is nil, so there is an such that . Thus is nil.
Suppose that and are nilpotent. Then there are natural numbers and such that and . Therefore, . ∎
The sum of an arbitrary family of nil ideals is nil.
Let be a ring, and let be a family of nil ideals of . Let . We must show that there is an with for every . Now, any such is actually in a sum of only finitely many of the ideals in . So it suffices to prove the lemma in the case that is finite. By induction, it is enough to show that the sum of two nil ideals is nil.
Let and be nil ideals of a ring . Then , and , which is nil. So by the first lemma, is nil. ∎
The sum of a finite family of nilpotent left or right ideals is nilpotent.
We prove this for right ideals. Again, by induction, it suffices to prove it for the case of two right ideals.
Let and be nilpotent right ideals of a ring . Then there are natural numbers and such that and .
Let . Let be elements of . We may write for each , with and . If we expand the product we get a sum of terms of the form where each .
Consider one of these terms . Then by our choice of , it must contain at least of the ’s or at least of the ’s. Without loss of generality, assume the former. So there are indices with for each . For , define , and define . Since is a right ideal, .
This is true for all choices of the , and so . But this says that . ∎
|Title||properties of nil and nilpotent ideals|
|Date of creation||2013-03-22 14:12:54|
|Last modified on||2013-03-22 14:12:54|
|Last modified by||mclase (549)|