Definition:Boolean Ring

Definition

A boolean ring (or Boolean ring) is an algebraic structure $\left({S, \circ, *}\right)$ such that:

$(1):\quad$ $S$ is closed under both $\circ$ and $*$

$(2):\quad$ Both $\circ$ and $*$ are commutative

$(3):\quad$ Both $\circ$ and $*$ distribute over the other

$(4):\quad$ Both $\circ$ and $*$ have identities $e^\circ$ and $e^*$ respectively, where $e^\circ \ne e^*$

$(5):\quad$ $\forall a \in S: \exists a' \in S: a \circ a' = e^*, a * a' = e^\circ$

A boolean ring can also be considered as a mathematical system $\left\{{S, O, A}\right\}$ where $O = \left\{{\circ, *}\right\}$ and $A$ consists of the set of axioms $(1)$ to $(5)$ as defined above.

At first glance, a boolean ring looks like an ordinary ring, except with the double distributivity thing in it.

But note that, despite the fact that Operations of Boolean Ring are Associative, neither $\left({S, \circ}\right)$ nor $\left({S, *}\right)$ are actually groups.

Alternative Names

A boolean ring is also referred to by some sources as a boolean algebra.

Also see

• Principle of Duality of Boolean Rings
• Operations of Boolean Ring are Idempotent
• Operations of Boolean Ring are Associative
• Identity Elements of Boolean Ring are also Zeroes

Source of Name

This entry was named for George Boole.

Sources

• W.E. Deskins: Abstract Algebra (1964): $\S 1.5$