properties of nil and nilpotent ideals

Lemma 1.

Let AB be ideals of a ring R. If A is nil and B/A is nil, then B is nil. If A is nilpotentPlanetmathPlanetmath and B/A is nilpotent, then B is nilpotent.


Suppose that A and B/A are nil. Let xB. Then xnA for some n, since B/A is nil. But A is nil, so there is an m such that xnm=(xn)m=0. Thus B is nil.

Suppose that A and B/A are nilpotent. Then there are natural numbersMathworldPlanetmath n and m such that Am=0 and BnA. Therefore, Bnm=0. ∎

Lemma 2.

The sum of an arbitrary family of nil ideals is nil.


Let R be a ring, and let be a family of nil ideals of R. Let S=II. We must show that there is an n with xn=0 for every xS. Now, any such x 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 inductionMathworldPlanetmath, it is enough to show that the sum of two nil ideals is nil.

Let A and B be nil ideals of a ring R. Then AA+B, and A+B/AB/(AB), which is nil. So by the first lemma, A+B is nil. ∎

Lemma 3.

The sum of a finite family of nilpotent left or right idealsMathworldPlanetmathPlanetmath is nilpotent.


We prove this for right ideals. Again, by induction, it suffices to prove it for the case of two right ideals.

Let A and B be nilpotent right ideals of a ring R. Then there are natural numbers n and m such that An=0 and bm=0.

Let k=n+m-1. Let z1,z2,,zk be elements of A+B. We may write zi=ai+bi for each i, with aiA and biB. If we expand the productPlanetmathPlanetmath z1z2zk we get a sum of terms of the form x1x2xk where each xi{ai,bi}.

Consider one of these terms x1x2xk. Then by our choice of k, it must contain at least n of the ai’s or at least m of the bi’s. Without loss of generality, assume the former. So there are indices i1<i2<<in with xijA for each j. For 1jn-1, define yj=xijxij+1xij+1-1, and define yn=xinxin+1xk. Since A is a right ideal, yjA.

Then x1x2xk=x1x2xi1-1y1y2ynx1x2xi1-1An=0.

This is true for all choices of the xi, and so z1z2zk=0. But this says that (A+B)k=0. ∎

Title properties of nil and nilpotent ideals
Canonical name PropertiesOfNilAndNilpotentIdeals
Date of creation 2013-03-22 14:12:54
Last modified on 2013-03-22 14:12:54
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Result
Classification msc 16N40
Related topic KoetheConjecture
Related topic NilIsARadicalProperty