Definition:Ring (Abstract Algebra)

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This page is about ring in the context of abstract algebra. For other uses, see ring.

Definition

A ring $\struct {R, *, \circ}$ is a semiring in which $\struct {R, *}$ forms an abelian group.


That is, in addition to $\struct {R, *}$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse.


Ring Axioms

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.


Note that a ring is still a semiring (in fact, an additive semiring), so all properties of these structures also apply to a ring.


Addition

The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as ring addition.


The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.


Product

The distributive operation $\circ$ in $\struct {R, *, \circ}$ is known as the ring product.


Binding Priority

In order to simplify expressions involving both $+$ and $\circ$, it is the convention that ring product has a higher precedence than ring addition:

$a \circ b + c := \paren {a \circ b} + c$


Ring Less Zero

It is convenient to have a symbol for $R \setminus \set 0$, that is, the set of all elements of the ring without the zero.

Thus we usually use:

$R_{\ne 0} = R \setminus \set 0$


Also defined as

Some sources insist on another criterion which a semiring $\struct {S, *, \circ}$ must satisfy to be classified as a ring:

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

Such sources then use the term rng (pronounced rung): for a ring without an identity.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ defines a ring as any structure fulfilling axioms $\text A 0$ - $\text A 4$, $\text M 0$ - $\text M 1$ and $\text D$, whether or not it has a unity.

The more specific structure which does have a unity is termed a ring with unity.


Other sources define a ring as an algebraic structure $\struct {R, *, \circ}$ which, while fulfilling all the other ring axioms, does not insist on $\text M 1$, associativity of ring product.

Such regimes refer to a ring which does fulfil axioms $\text A 0$ - $\text A 4$, $\text M 0$ - $\text M 1$, $\text D$ as an associative ring.


Also known as

Earlier sources, that is, dating to the early $20$th century, refer to a ring as an annulus, but the word ring (at least in this context) is now generally ubiquitous.


Also see


Categories of Rings

  • If $\struct {R^*, \circ}$ is a group, then $\struct {R, +, \circ}$ is called a division ring.
  • If $\struct {R^*, \circ}$ is an abelian group, then $\struct {R, +, \circ}$ is called a field.


Generalizations



Historical Note

According to Ian Stewart, in his Galois Theory, 3rd ed. of $2004$, the ring axioms were first formulated by Heinrich Martin Weber in $1893$.


Internationalization

Ring is translated:

In French : anneau
In German : Ring  (that is: ${}$a contraction of its original name Zahlring, which means ring of integers)


Sources

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