In the definition of a ring $R$, one has
- $a(b+c) = ab + ac$ and
- $(a+b)c = ac + bc$
for all $a,b,c\in R$
My question is (just out of curiosity) if one really needs both of these. I can't think of an example of something that is not a ring that only satisfies one of the sides of the distributive law. So can one prove that if $a(b+c) = ab + ac$ for all $a,b,c$, then $(a+b)c = ac + bc$ for all $a,b,c$.
Edit: I maybe should add that all rings in my definition have a unity $1$.