Definition:Ringoid (Abstract Algebra)

Definition
A ringoid is a triple $\left({S, *, \circ}\right)$ where:
 * $S$ is a set
 * $*$ and $\circ$ are binary operations on $S$
 * the operation $\circ$ distributes over $*$.

That is:


 * $\forall a, b, c \in S: a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$
 * $\forall a, b, c \in S: \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right)$

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

Similarly, for $\left({a \circ b}\right) * \left({a \circ c}\right)$ 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, $\left({S, *, \circ}\right)$, 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.

Special cases

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