# Definition:Boolean Ring

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

## Contents

## 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 | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b \in R \) | ||||

\((\text A 1)\) | $:$ | Associativity of addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \) | ||||

\((\text A 2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b = b * a \) | ||||

\((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\displaystyle \exists 0_R \in R: \forall a \in R:\) | \(\displaystyle a * 0_R = a = 0_R * a \) | ||||

\((\text {AC} 2)\) | $:$ | Characteristic 2 for addition: | \(\displaystyle \forall a \in R:\) | \(\displaystyle a * a = 0_R \) | ||||

\((\text M 0)\) | $:$ | Closure under product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b \in R \) | ||||

\((\text M 1)\) | $:$ | Associativity of product | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||

\((\text M 2)\) | $:$ | Identity element for product: the unity | \(\displaystyle \exists 1_R \in R: \forall a \in R:\) | \(\displaystyle 1_R \circ a = a = a \circ 1_R \) | ||||

\((\text {MI})\) | $:$ | Idempotence of product | \(\displaystyle \forall a \in R:\) | \(\displaystyle a \circ a = a \) | ||||

\((\text D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \) | ||||

\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |

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.

## 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): $\S 1$