# The sum of a nilpotent left ideal and a nil left ideal

In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.

I am having trouble with this. Can anyone help?

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Very nice problem! But as it seems to be homework, I'll just give hints. Let $a$ be in a nil ideal (thus nilpotent, say $a^n=0$), and $b$ be in a nilpotent ideal of order $m$. Expand $\left(a+b\right)^u$ for a big $u$ (much bigger than the $n+m-1$ you hopefully know from commutative algebra). Every addend which has enough $b$'s is $0$ (why?). Now if $u$ is chosen big enough, what can you tell about the remaining addends? – darij grinberg Mar 15 '11 at 21:36
@Darij: It is a very bad style answering homework problems on MO (or even giving a hint). I voted to close. – Mark Sapir Mar 15 '11 at 21:42
By the way, it is unknown if the sum of two left nil ideals is nil. en.wikipedia.org/wiki/K%C3%B6the_conjecture – Martin Brandenburg Mar 15 '11 at 22:20