"-1" is defined, in a ring, as the additive inverse of the multiplicative identity. In particular, 1+ (-1)= 0 so, by the distributive law, . Because 1 is the multiplicative identity, , of course, and so that . Since -1 is the additive inverse of 1, and this equation says it is the additive inverse of , it follows that .

A 'unit' in a ring is a member that has a multiplicative inverse. If u is a unit, there exist "v" such that uv= vu= 1. But we have also that (-u)(-v)= uv= 1 (use the distributive law to prove that) so that the multiplicative inverse of -u is -v.