Ring problem: nilpotent elements and units

[rachellcb]

Homework Statement

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

The Attempt at a Solution

Need to show that u'(u + a₁ + ... + a_k)=1 for some u' $\in$ R.
I know that 1 - a_i is a unit (i=1,2,...,k) since each a_i is nilpotent.
So I'm trying to think of possible candidates for u'.
My attempt so far have been to try and cancel the a_i terms by multiplying
u + a₁ + ... + a_k by a₁^n-1, where a₁ⁿ =0 and so on for all all a_i but this leaves me with ua₁^n-1...a_k^m-1 and doesn't seem to be leading anywhere...
Any suggestions?

 

[jgens]

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).

 

[rachellcb]

Solved, thanks!