Definition:Boolean Ring

This page is about Boolean Ring in the context of Abstract Algebra. For other uses, see Ring.

Definition

Let $\struct {R, +, \circ}$ be a ring.

Then $R$ is called a Boolean ring if and only if $R$ is an idempotent ring with unity.

Boolean Ring Axioms

More abstractly, a Boolean ring is an algebraic structure $\struct {R, *, \circ}$ subject to 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	\(\ds \forall a, b, c \in R:\)	\(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \)
					\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)

These criteria are called the Boolean ring axioms.

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Also defined as

Some sources use the (deprecated) name Boolean ring to mean what is better known as a Boolean algebra.

Others define it simply to mean what we have called an idempotent ring, not imposing that it have a unity.

Also see

• Definition:Boolean Algebra

• Results about Boolean rings can be found here.

Source of Name

This entry was named for George Boole.

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$