Definition:Boolean Ring

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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.


Sources