# Definition:Boolean Ring

This page is about boolean rings in the context of abstract algebra. For other uses, see Definition:Ring.

## Definition

Let $\left({R, +, \circ}\right)$ 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 $\left({R, *, \circ}\right)$ subject to the following axioms:

 $(A0)$ $:$ Closure under addition $\displaystyle \forall a, b \in R:$ $\displaystyle a * b \in R$ $(A1)$ $:$ Associativity of addition $\displaystyle \forall a, b, c \in R:$ $\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)$ $(A2)$ $:$ Commutativity of addition $\displaystyle \forall a, b \in R:$ $\displaystyle a * b = b * a$ $(A3)$ $:$ Identity element for addition: the zero $\displaystyle \exists 0_R \in R: \forall a \in R:$ $\displaystyle a * 0_R = a = 0_R * a$ $(AC2)$ $:$ Characteristic 2 for addition: $\displaystyle \forall a \in R:$ $\displaystyle a * a = 0_R$ $(M0)$ $:$ Closure under product $\displaystyle \forall a, b \in R:$ $\displaystyle a \circ b \in R$ $(M1)$ $:$ Associativity of product $\displaystyle \forall a, b, c \in R:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ $(M2)$ $:$ Identity element for product: the unity $\displaystyle \exists 1_R \in R: \forall a \in R:$ $\displaystyle 1_R \circ a = a = a \circ 1_R$ $(MI)$ $:$ Idempotence of product $\displaystyle \forall a \in R:$ $\displaystyle a \circ a = a$ $(D)$ $:$ Product is distributive over addition $\displaystyle \forall a, b, c \in R:$ $\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right),$ $\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right)$

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.