# 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.

## 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

• Results about Boolean rings can be found here.

## Source of Name

This entry was named for George Boole.