\begin{definition}[Definition:Commutative and Unitary Ring/Axioms]
A commutative and unitary ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:
{{begin-axiom}}
{{axiom | n = \text A 0
        | q = \forall a, b \in R
        | m = a * b \in R
        | lc= Closure under addition
}}
{{axiom | n = \text A 1
        | q = \forall a, b, c \in R
        | m = \paren {a * b} * c = a * \paren {b * c}
        | lc= Associativity of addition
}}
{{axiom | n = \text A 2
        | q = \forall a, b \in R
        | m = a * b = b * a
        | lc= Commutativity of addition
}}
{{axiom | n = \text A 3
        | q = \exists 0_R \in R: \forall a \in R
        | m = a * 0_R = a = 0_R * a
        | lc= Identity element for addition: the zero
}}
{{axiom | n = \text A 4
        | q = \forall a \in R: \exists a' \in R
        | m = a * a' = 0_R = a' * a
        | lc= Inverse elements for addition: negative elements
}}
{{axiom | n = \text M 0
        | q = \forall a, b \in R
        | m = a \circ b \in R
        | lc= Closure under product
}}
{{axiom | n = \text M 1
        | q = \forall a, b, c \in R
        | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}
        | lc= Associativity of product
}}
{{axiom | n = \text M 2
        | q = \forall a, b \in R
        | m = a \circ b = b \circ a
        | lc= Commutativity of product
}}
{{axiom | n = \text M 3
        | q = \exists 1_R \in R: \forall a \in R
        | m = a \circ 1_R = a = 1_R \circ a
        | lc= Identity element for product: the unity
}}
{{axiom | n = \text D
        | q = \forall a, b, c \in R
        | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}
        | lc= Product is distributive over addition
}}
{{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}
}}
{{end-axiom}}
These criteria are called the '''commutative and unitary ring axioms'''.
These can be alternatively presented as:
{{begin-axiom}}
{{axiom | n = \text A
        | t = $\struct {R, *}$ is an abelian group
}}
{{axiom | n = \text M
        | t = $\struct {R, \circ}$ is a commutative monoid
}}
{{axiom | n = \text D
        | t = $\circ$ distributes over $*$
}}
{{end-axiom}}
\end{definition}