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.