In a unital ring, the negative of a unit is a unit

Let R be a ring with 1. Prove that if u is a unit in R, then so is -u.


Since u is a unit, we have uv = vu = 1 for some v \in R. Thus we have (-v)(-u) = vu = 1 and (-u)(-v) = uv = 1. Thus -u is a unit.

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