Ring problem: nilpotent elements and units

Homework Statement

Let R be a ring with multiplicative identity. Let u $\in$ R be a unit and let a1, ..., ak be nilpotent elements that commute with each other and with u. To show: u + a1 + ... + ak is a unit.

The Attempt at a Solution

Need to show that u'(u + a1 + ... + ak)=1 for some u' $\in$ R.
I know that 1 - ai is a unit (i=1,2,...,k) since each ai is nilpotent.
So i'm trying to think of possible candidates for u'.
My attempt so far have been to try and cancel the ai terms by multiplying
u + a1 + ... + ak by a1n-1, where a1n =0 and so on for all all a_i but this leaves me with ua1n-1...akm-1 and doesn't seem to be leading anywhere...
Any suggestions?

Answers and Replies

jgens
Gold Member
Show first that the sum of two nilpotent elements which commute with each other is nilpotent (Hint: Generalize the binomial theorem to an arbitrary ring). Then use induction to show that the sum of k nilpotent elements which commute with each other is nilpotent. Then lastly prove that the sum of a nilpotent element and a unit is a unit (Hint: This is similar to proving 1+x is a unit, where x is nilpotent).

Solved, thanks!