Axiom:Boolean Ring Axioms

Definition

A Boolean ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, satisfying the following conditions:

$(\text A 0)$	$:$	Closure under addition	$\ds \forall a, b \in R:$	$\ds a * b \in R $
$(\text A 1)$	$:$	Associativity of addition	$\ds \forall a, b, c \in R:$	$\ds \paren {a * b} * c = a * \paren {b * c} $
$(\text A 2)$	$:$	Commutativity of addition	$\ds \forall a, b \in R:$	$\ds a * b = b * a $
$(\text A 3)$	$:$	Identity element for addition: the zero	$\ds \exists 0_R \in R: \forall a \in R:$	$\ds a * 0_R = a = 0_R * a $
$(\text {AC} 2)$	$:$	Characteristic 2 for addition:	$\ds \forall a \in R:$	$\ds a * a = 0_R $
$(\text M 0)$	$:$	Closure under product	$\ds \forall a, b \in R:$	$\ds a \circ b \in R $
$(\text M 1)$	$:$	Associativity of product	$\ds \forall a, b, c \in R:$	$\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} $
$(\text M 2)$	$:$	Identity element for product: the unity	$\ds \exists 1_R \in R: \forall a \in R:$	$\ds 1_R \circ a = a = a \circ 1_R $
$(\text {MI})$	$:$	Idempotence of product	$\ds \forall a \in R:$	$\ds a \circ a = a $
$(\text D)$	$:$	Product is distributive over addition	$\ds \forall a, b, c \in R:$	$\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, $
				$\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} $

These criteria are called the Boolean ring axioms.

\((\text A 0)\)	$:$	Closure under addition	\(\ds \forall a, b \in R:\)	\(\ds a * b \in R \)
\((\text A 1)\)	$:$	Associativity of addition	\(\ds \forall a, b, c \in R:\)	\(\ds \paren {a * b} * c = a * \paren {b * c} \)
\((\text A 2)\)	$:$	Commutativity of addition	\(\ds \forall a, b \in R:\)	\(\ds a * b = b * a \)
\((\text A 3)\)	$:$	Identity element for addition: the zero	\(\ds \exists 0_R \in R: \forall a \in R:\)	\(\ds a * 0_R = a = 0_R * a \)
\((\text {AC} 2)\)	$:$	Characteristic 2 for addition:	\(\ds \forall a \in R:\)	\(\ds a * a = 0_R \)
\((\text M 0)\)	$:$	Closure under product	\(\ds \forall a, b \in R:\)	\(\ds a \circ b \in R \)
\((\text M 1)\)	$:$	Associativity of product	\(\ds \forall a, b, c \in R:\)	\(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)
\((\text M 2)\)	$:$	Identity element for product: the unity	\(\ds \exists 1_R \in R: \forall a \in R:\)	\(\ds 1_R \circ a = a = a \circ 1_R \)
\((\text {MI})\)	$:$	Idempotence of product	\(\ds \forall a \in R:\)	\(\ds a \circ a = a \)
\((\text D)\)	$:$	Product is distributive over addition	\(\ds \forall a, b, c \in R:\)	\(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \)
				\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)