Definition:Commutative and Unitary Ring/Axioms

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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:

\((\text A 0)\)   $:$   Closure under addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b \in R \)             
\((\text A 1)\)   $:$   Associativity of addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \)             
\((\text A 2)\)   $:$   Commutativity of addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b = b * a \)             
\((\text A 3)\)   $:$   Identity element for addition: the zero      \(\displaystyle \exists 0_R \in R: \forall a \in R:\) \(\displaystyle a * 0_R = a = 0_R * a \)             
\((\text A 4)\)   $:$   Inverse elements for addition: negative elements      \(\displaystyle \forall a \in R: \exists a' \in R:\) \(\displaystyle a * a' = 0_R = a' * a \)             
\((\text M 0)\)   $:$   Closure under product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b \in R \)             
\((\text M 1)\)   $:$   Associativity of product      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)             
\((\text M 2)\)   $:$   Commutativity of product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b = b \circ a \)             
\((\text M 3)\)   $:$   Identity element for product: the unity      \(\displaystyle \exists 1_R \in R: \forall a \in R:\) \(\displaystyle a \circ 1_R = a = 1_R \circ a \)             
\((\text D)\)   $:$   Product is distributive over addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)             
\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)             

These criteria are called the commutative and unitary ring axioms.

These can be alternatively presented as:

\((\text A)\)   $:$   $\struct {R, *}$ is an abelian group             
\((\text M)\)   $:$   $\struct {R, \circ}$ is a commutative monoid             
\((\text D)\)   $:$   $\circ$ distributes over $*$             

Also see