Exercise 3.9

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

Answer

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

$$\begin{array}{r}(-a)\cdot b+a\cdot b=(-a+a)\cdot b=0\cdot b=0\end{array}$$

Thus, $(-a)\cdot b=-a\cdot b$. Similarly,

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

Thus, $a\cdot (-b)=-a\cdot b$. Therefore, $(-a)\cdot b=a\cdot (-b)=-a\cdot b$. $■$