Exercise 3.6

Let $R$ be a ring. Prove that $\forall a\in R,(-1)\cdot a=-a$.

Answer

Let $R$ be a ring, and $a\in R$. Then,

$$0=0\cdot a=(1+(-1))\cdot a=1\cdot a+(-1)\cdot a=a+(-1)\cdot a$$

Since multiplicative inverses are unique, then $(-1)\cdot a=-a$. $■$