Symmetric Difference with Intersection forms Ring/Proof 1

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Theorem

Let $S$ be a set.


Then $\left({\mathcal P \left({S}\right), *, \cap}\right)$ is a commutative ring with unity, in which the unity is $S$.


This ring is not an integral domain.


Proof

From Symmetric Difference on Power Set forms Abelian Group, $\struct {\powerset S, *}$ is an abelian group, where $\O$ is the identity and each element is self-inverse.

From Power Set with Intersection is Monoid, $\struct {\powerset S, \cap}$ is a commutative monoid whose identity is $S$.

Also Intersection Distributes over Symmetric Difference.

Thus $\struct {\powerset S, \cap}$ is a commutative ring with a unity which is $S$.

From Intersection with Empty Set:

$\forall A \in \powerset S: A \cap \O = \O = \O \cap A$

Thus $\O$ is indeed the zero.


However, from Set Intersection Not Cancellable, it follows that $\struct {\powerset S, *, \cap}$ is not an integral domain.

$\blacksquare$


Sources