Definition:Ringoid (Abstract Algebra)

Definition

A ringoid is a triple $\struct {S, *, \circ}$ where:

$S$ is a set

$*$ and $\circ$ are binary operations on $S$

the operation $\circ$ distributes over $*$.

That is:

$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$

$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$

Closedness

For the expression $a \circ \paren {b * c}$ to make sense, we require that $S$ is closed under $*$.

Similarly, for $\paren {a \circ b} * \paren {a \circ c}$ to make sense, we require that $S$ is closed under $\circ$.

Note that $\circ$ does not have to be associative.

Note on order of operations

In the denotation of this structure, $\struct {S, *, \circ}$, the distributor is shown after the distributand.

In the context of a ringoid, the fact that $\circ$ distributes over $*$ is known as the distributive law.

Also see

Stronger properties

• Definition:Semiring (Abstract Algebra)
• Definition:Ring (Abstract Algebra)

Sources

• Weisstein, Eric W. "Ringoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ringoid.html