# Definition:Boolean Ring

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

## Contents

## 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.

## Sources

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$