Minus times Minus is Plus in a Ring

For $a,b$ in a ring, $(-a)(-b)=ab$ is not an extra sign convention. It follows from distributivity and the uniqueness of additive inverses. Here $-a$ means the element cancelling $a$, not necessarily a negative number.

Multiplication preserves zero

Distributivity gives $a0=a(0+0)=a0+a0$. Cancelling $a0$ gives $a0=0$; similarly, $0b=0$.

Each minus sign forces a cancellation

First, multiply $a+(-a)=0$ by $b$:

$$ab+(-a)b=(a+(-a))b=0.$$

Thus $(-a)b$ cancels $ab$, so $(-a)b=-(ab)$.

Next, multiply $b+(-b)=0$ by $-a$ and substitute:

$$\begin{array}{rl}(-a)b+(-a)(-b)& =(-a)(b+(-b))=0,\\ -(ab)+\underset{\text{must cancel }-(ab)}{\underbrace{(-a)(-b)}}& =0.\end{array}$$

Since $ab$ already cancels $-(ab)$, uniqueness forces

$$\boxed{(-a)(-b)=ab.}$$

No division, multiplicative commutativity or multiplicative identity was needed; a field is only a special case. “Plus” means the original product, not positivity. The sign rule is a consequence of preserving cancellation.