In an arbitrary ring, I have that $ab = 1$. I'm pretty certain that $a(-b) = -1$, but I'm not sure how to prove that the $-$ sign commutes. Is this necessarily true? Could someone point me in the direction of a proof or counterexample?
2026-03-30 06:22:05.1774851725
Showing that multiplying by $-1$ commutes with a unit and its inverse in an arbitrary ring.
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Hint: You want to prove $a(-b)=-(ab)$. That is, you want to prove $a(-b)$ is the additive inverse of $ab$. So try adding $a(-b)$ and $ab$ and see if you can show the sum is $0$.