Definition:Ringoid (Abstract Algebra)

Definition
A ringoid $$\left({S, *, \circ}\right)$$ is a set $$S$$ on which is defined two binary operations, here denoted $$\circ$$ and $$*$$, defined on all the elements of $$S \times S$$, where one operation $$\circ$$ distributes over the other $$*$$.

That is:


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

Note that in a ringoid, there is no insistence that $$S$$ is closed under either of these operations, neither does $$\circ$$ have to be associative.

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.