# Definition:Ring (Abstract Algebra)

## 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 $\displaystyle \forall a, b \in R:$ $\displaystyle a * b \in R$ $(\text A 1)$ $:$ 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 $\displaystyle \forall a, b, c \in R:$ $\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c}$ $(\text D)$ $:$ 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 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.

The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as ring addition, or just 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 $\displaystyle \exists 1_R \in R: \forall a \in R:$ $\displaystyle a \circ 1_R = a = 1_R \circ a$

Such sources refer to what this website calls a ring as a rng (pronounced rung): that is a ring without an identity.

However, this website specifically defines a ring as one fulfilling axioms $\text A 0$ - $\text A$, $\text M 0$ - $\text M 1$, $\text D$ only, and instead refers to this more specific structure as 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, tend to use the term annulus for this concept, but the word ring (at least in this context) is now generally ubiquitous.

## Also see

• 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.
• Results about rings can be found here.

## 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 $1893$.