Disjunction as sum operation in Boolean Ring Boolean ring is defined with operations of ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). I understand that algebraic structure involving conjunction and disjunction is lattice. Yet, I struggle to witness what ring axiom is violated if we take disjunction as ring addition.

If we take disjunction for addition then the ring axiom "for every $x$ there exists $y$ such that $x+y=0$" is violated.
Looking at things as a lattice, if $x>0$ then $x\lor y\ge x >0$, so $x+y\ne 0$.
Or to be concrete, consider the power set of $X$. Given $A\subset X$ with $A\ne\emptyset$, there is no $B\subset X$ with $A\cup B=\emptyset$.
