Boolean Rings

Boolean Rings are an Algebra $(S,\otimes ,\wedge )$ with two binary operations which satisfy the following axioms:

Axioms

#	Name	FOL
1.	Closure under addition	$\forall a,b\in S:a\otimes b\in S$
2.	Associativity under addition	$\forall a,b\in S:(a\otimes b)\otimes c=a\otimes (b\otimes c)$
3.	Commutativity under addition	$\forall a,b\in S:a\otimes b=b\otimes a$
4.	Identity element under addition	$\exists 0\in S:\forall a\in S:a\otimes 0=a=0\otimes a$
5.	Closure under product	$\forall a,b\in S:a\wedge b\in S$
6.	Associativity under product	$\forall a,b\in S:(a\wedge b)\wedge c=a\wedge (b\wedge c)$
7.	Identity element for product	$\exists 1\in S:\forall a\in S:a\wedge 1=a=1\wedge a$
8.	Idempotence of product	$\forall a\in S:a\wedge a=a$
9.	distributive property of product over addition	$\forall a,b,c\in S:a\wedge (b\otimes c)=a\wedge b\otimes a\wedge c$

You will note that the thing which distinguishes a boolean ring from others is idempotence of the product.