## 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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