# Axiom:Ring Axioms

## Definition

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

These criteria are called the ring axioms.

## Also presented as

These can also be presented as:

 $(\text A)$ $:$ $\struct {R, *}$ is an abelian group $(\text M 0)$ $:$ $\struct {R, \circ}$ is closed $(\text M 1)$ $:$ $\circ$ is associative on $R$ $(\text D)$ $:$ $\circ$ distributes over $*$

## Also defined as

For a ring with unity, the following axiom also holds:

 $(\text M 2)$ $:$ Identity element for $\circ$: the unity $\ds \exists 1_R \in R: \forall a \in R:$ $\ds a \circ 1_R = a = 1_R \circ a$

For a commutative ring, the following axiom also holds:

 $(\text C)$ $:$ Commutativity of Ring Product $\ds \forall a, b \in R:$ $\ds a \circ b = b \circ a$