A Boolean ring is a ring which all of its elements are idempotent, i.e. $a ^{2}=a$. I know that If we interpret multiplication and addition in such a ring, as meet and joint respectively, then Boolean Ring is essentially the same as Boolean Algebra. But, I don't know how to show that multiplication can be distributed over addition in a Boolean Ring,
$$ a+\left(b.c\right)=\left(a+b\right).\left(a+c\right) $$ a property which is an axiom in Boolean Algebra; $$ a \vee \left(b \wedge c \right)=\left(a\vee b\right)\wedge\left(a\vee c\right) $$
Nooo.. Addition is not the join operation, it is rather the symmetric difference: $$ a+b := (a\land\lnot b) \lor (\lnot a\land b)$$