\begin{definition}[Definition:Rig]
A '''rig''' is an additive semiring $\struct {S, *, \circ}$ in which $\struct {S, *}$ is a monoid.
Alternatively, this is a semiring in which $\struct {S, *}$ is a commutative monoid.
That is, $\struct {S, *, \circ}$ has the following properties:
{{begin-axiom}}
{{axiom | n = \text A 0
        | q = \forall a, b \in S
        | m = a * b \in S
        | lc= Closure under $*$
}}
{{axiom | n = \text A 1
        | q = \forall a, b, c \in S
        | m = \paren {a * b} * c = a * \paren {b * c}
        | lc= Associativity of $*$
}}
{{axiom | n = \text A 2
        | q = \forall a, b \in S
        | m = a * b = b * a
        | lc= Commutativity of $*$
}}
{{axiom | n = \text A 3
        | q = \exists 0_S \in S: \forall a \in F
        | m = a * 0_S = a = 0_S * a
        | lc= Identity element for $*$: the zero
}}
{{axiom | n = \text M 0
        | q = \forall a, b \in S
        | m = a \circ b \in S
        | lc= Closure under $\circ$
}}
{{axiom | n = \text M 1
        | q = \forall a, b, c \in S
        | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}
        | lc= Associativity of $\circ$
}}
{{axiom | n = \text M 2
        | q = \forall a \in S
        | m = a \circ 0_S = 0_S = 0_S \circ a
        | lc= The zero is a zero element for $\circ$
}}
{{axiom | n = \text D
        | q = \forall a, b, c \in S
        | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c}
        | lc= $\circ$ is distributive over $*$
}}
{{end-axiom}}
Note that the zero element needs to be specified here as an axiom: $\text M 2$.
By Ring Product with Zero, in a ring, the property $\text M 2$ of the zero element follows as a consequence of the ring axioms.
\end{definition}