Ring/Basic properties/Fact

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

${}a0=0$ (annihilation rule).
${}(-a)b=-ab=a(-b)\,$

(rules for sign).
${}(-a)(-b)=ab\,.$
${}a(b-c)=ab-ac\,.$
${}{\left(\sum _{i=1}^{r}a_{i}\right)}{\left(\sum _{k=1}^{s}b_{k}\right)}=\sum _{1\leq i\leq r,\,1\leq k\leq s}a_{i}b_{k}$ (general law of distributivity).

Proof, Write another proof