# 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?