Definition:Boolean Ring

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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      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b \in R \)             
\((\text A 1)\)   $:$   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