- #1

rachellcb

- 10

- 0

## Homework Statement

Let R be a ring with multiplicative identity. Let u [itex]\in[/itex] R be a unit and let a

_{1}, ..., a

_{k}be nilpotent elements that commute with each other and with u. To show: u + a

_{1}+ ... + a

_{k}is a unit.

## The Attempt at a Solution

Need to show that u'(u + a

_{1}+ ... + a

_{k})=1 for some u' [itex]\in[/itex] 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 + a

_{1}+ ... + a

_{k}by a

_{1}

^{n-1}, where a

_{1}

^{n}=0 and so on for all all a_i but this leaves me with ua

_{1}

^{n-1}...a

_{k}

^{m-1}and doesn't seem to be leading anywhere...

Any suggestions?