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

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.2

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

Solution: Since $u$ is a unit, we have $uv = vu = 1$ for some $v \in R$. Thus by Exercise 7.1.1, we have $$(-v)(-u) = vu = 1$$ and $$(-u)(-v) = uv = 1.$$ Thus $-u$ is a unit.


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