Field/Basic properties/Fact

Let $K$ be a field, and let $a, b, c, a_{1}, \dots, a_{r}, b_{1}, \dots, b_{s}$ denote elements from $K$ . Then the following statements hold.

$a 0 = 0$ (annulation rule).
$(- a) b = - a b = a (- b)$

(rules for sign).
$(- a) (- b) = a b .$
$a (b - c) = a b - a c .$
From $a \cdot b = 0$ one can deduce $a = 0$ or $b = 0$ .
$(\sum_{i = 1}^{r} a_{i}) (\sum_{k = 1}^{s} b_{k}) = \sum_{1 \leq i \leq r, 1 \leq k \leq s} a_{i} b_{k}$ (general law of distributivity).