Definition:Ring (Abstract Algebra)

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

Definition

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

That is, in addition to $\left({R, *}\right)$ being closed and associative 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 is defined two binary operations $\circ$ and $*$, which satisfies the following conditions:

A:	$\left({R, *}\right)$ is a group
M0:	$\left({R, \circ}\right)$ is closed
M1:	$\circ$ is associative on $R$
D:	$\circ$ distributes over $*$.

These four stipulations are called the ring axioms.

Note that a ring is still a semiring, so all properties of a semiring also apply to a ring.

Ring Product

The distributive operation $\circ$ in $\left({R, *, \circ}\right)$ is known as the ring product.

Binding Priority

We usually simplify our brackets somewhat, by imposing the rule:

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

... that is, ring product has a higher precedence than addition.

Element Categories

The elements in a ring are partitioned into three classes:

1. the zero
2. the units
3. the proper elements.

Ring Less Zero

It is convenient to have a symbol for $R \setminus \left\{{0}\right\}$, that is, the set of all elements of the ring without the zero. Thus we usually use:

$R^* = R \setminus \left\{{0}\right\}$

Historical Note

According to Ian Stewart, the ring axioms were first formulated by Heinrich Martin Weber in 1893.

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 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.

• If $\left({R^*, \circ}\right)$ is a group, then $\left({R, +, \circ}\right)$ is a division ring.

• If $\left({R^*, \circ}\right)$ is a abelian group, then $\left({R, +, \circ}\right)$ is a field.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.3$: Footnote
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.18$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$: Definitions $1.1 \ \text{(c)}$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 54$

• Ring. O.A. Ivanova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php/Ring