# Definition:Ring (Abstract Algebra)

*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**, 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 | \(\ds \exists 1_R \in R: \forall a \in R:\) | \(\ds a \circ 1_R = a = 1_R \circ a \) |

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

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

- A commutative ring is a ring $\struct {R, +, \circ}$ in which the ring product $\circ$ is commutative.

- If $\struct {R^*, \circ}$ is a monoid, then $\struct {R, +, \circ}$ is called a ring with unity.

- A commutative and unitary ring is a commutative ring $\struct {R, +, \circ}$ which at the same time is a ring with unity.

- An integral domain is a commutative and unitary ring which has no proper zero divisors.

- 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.

### 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 $1893$.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain: Footnote - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 18$. Definition of a Ring - 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 - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**ring**

- Weisstein, Eric W. "Ring." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Ring.html - Ring. O.A. Ivanova (originator),
*Encyclopedia of Mathematics*. URL: https://www.encyclopediaofmath.org/index.php?title=Ring