Definition:Ringoid (Abstract Algebra)
Jump to navigation
Jump to search
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
Sources
- Weisstein, Eric W. "Ringoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ringoid.html