Ring of Sets vs Ring in Universal Algebra
Ring of Sets by corresponding union to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well) satisfy most ring axioms. However the fundamental difference is that it doesn't have an additive inverse because $X\cup Y=\varnothing$ if and only if $X=\varnothing$ and $Y=\varnothing$.
Thus your definition of Ring of Sets is not a ring rigorously.
Another way to define Ring of Sets is known as Boolean Ring, i.e by corresponding symmetric difference of sets to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well). It satisfies all ring axioms with an additive inverse.
Please read this post for more detail.