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.

Stronger properties

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