Boolean algebra-Boolean ring. Stone Theorem?

Question

I am interested in knowing which theorem is responsible for the following statement:

Every Boolean algebra can become a Boolean ring by taking the ring addition to be $A\oplus B = (A \land \lnot B) \lor (\lnot A \land B)$ and the ring multiplication to be $A\odot B = A \land B$.

In which way are sigma ideals a special case of ideals?

Is it Stone Theorem?

Answer 1

Boolean rings are rings with unit, and with all of its elements idempotent:

$x^2 = x$

Boolean algebras can be regarded as boolean rings through the definitions you have given for product and sum. Also boolean rings can be regarded as boolean algebras through the following definitions of join, meet and complement:

$A \lor B = A \oplus B \oplus (A\otimes B)$

$A \land B = A \odot B$

$A' = A \oplus 1$

All this is basic theory and has nothing to do with some named theorem.