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.

These can also be presented 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:

$(\text C)$

$:$

$\ds \forall a, b \in R:$

$\ds a \circ b = b \circ a $

1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences

\((\text A 0)\)	$:$	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	\(\ds \forall a, b, c \in R:\)	\(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)
				\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)