1. The problem statement, all variables and given/known data Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R. 2. Relevant equations A multiplicative inverse if (1+r)*x = 1 where x is some element in R. 3. 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?