- #1

RJLiberator

Gold Member

- 1,095

- 63

## Homework Statement

Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R.

## Homework Equations

A multiplicative inverse if (1+r)*x = 1 where x is some element in R.

## The Attempt at a Solution

We know we have to use two facts.

1. Multiplicative inverse (1+r)*x=1

2. the fact that r^2 = 0.

So I first tried, well, why don't we use (1+r)(1+r) = 0

However, this only got me to 1+r^2+2r = 0 with r^2 =0 we have 1+2r = 0

But that doesn't appear to be what we wanted to show.

Althought, we could subtract -1 from both sides, and subtract r from both sides to get

r = -1-r

r = -(1+r)

But this still isn't what we wanted to show.

If I divide both sides by r, then I get 1 = -1/r (1+r) and this would work, however, I can't assume element r can be divided to get 1. (at least I dont think I can based off the few multiplicative axioms I have for rings/fields).

I have to think that I am on the right path, but any hints to get me closer?