# Definition:Ring (Abstract Algebra)

*This page is about rings in the context of abstract algebra. For other uses, see Definition:Ring.*

## Contents

## Definition

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

That is, in addition to $\left({R, *}\right)$ 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 $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((A0)\) | $:$ | Closure under addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b \in R \) | ||||

\((A1)\) | $:$ | Associativity of addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \) | ||||

\((A2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b = b * a \) | ||||

\((A3)\) | $:$ | Identity element for addition: the zero | \(\displaystyle \exists 0_R \in R: \forall a \in R:\) | \(\displaystyle a * 0_R = a = 0_R * a \) | ||||

\((A4)\) | $:$ | Inverse elements for addition: negative elements | \(\displaystyle \forall a \in R: \exists a' \in R:\) | \(\displaystyle a * a' = 0_R = a' * a \) | ||||

\((M0)\) | $:$ | Closure under product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b \in R \) | ||||

\((M1)\) | $:$ | Associativity of product | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \) | ||||

\((D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right) \) | ||||

\(\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right) \) |

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 $\left({R, *, \circ}\right)$ is referred to as **ring addition**, or just **addition**.

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

## Additive Group

The group $\struct {R, +}$ is known as the **additive group of $R$**.

### Ring Product

The distributive operation $\circ$ in $\left({R, *, \circ}\right)$ 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:

- $a \circ b + c := \left({a \circ b}\right) + c$

### 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 $\left({S, *, \circ}\right)$ must satisfy to be classified as a **ring**:

\((M2):\quad\) | \(\displaystyle \exists 1_R \in R: \forall a \in R:\) | \(\) | \(\displaystyle a \circ 1_R = a = 1_R \circ a\) | $\quad$ Identity element for $\circ$: the unity | $\quad$ |

Such sources refer to what this website calls a **ring** as a **rng** (pronounced "rung"): i.e. a "ring" without an "identity".

However, this website specifically defines a **ring** as one fulfilling axioms $A0 - A4, M0 - M1, D$ only, and instead refers to this more specific structure as a ring with unity.

Other sources define a **ring** as an algebraic structure $\left({R, *, \circ}\right)$ which, while fulfilling all the other ring axioms, does not insist on $M1$, associativity of ring product.

Such regimes refer to a **ring** which does fulfil axioms $A0 - A4, M0 - M1, D$ as an **associative ring**.

## Also see

- A commutative ring is a ring $\left({R, +, \circ}\right)$ in which the ring product $\circ$ is commutative.

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

- A commutative and unitary ring is a commutative ring $\left({R, +, \circ}\right)$ 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 $\left({R^*, \circ}\right)$ is a group, then $\left({R, +, \circ}\right)$ is called a division ring.

- If $\left({R^*, \circ}\right)$ is an abelian group, then $\left({R, +, \circ}\right)$ 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): $\S 1.1$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 21$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 1.3$: Footnote - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 5.18$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 1.1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 54$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$

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