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Ring (mathematics)

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. The branch of mathematics where rings are studied is called ring theory.

Table of contents
1 History
2 Definition and notation
3 Examples
4 Simple theorems
5 Constructing new rings from given ones
6 Glossary and related topics


See Ring theory

Definition and notation

A ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,
a * (b*c'\') = (a*b) * c''
a * (b+c) = (a*b) + (a*c)
(a+b) * c = (a*c) + (b*c)

and such that there exists a multiplicative identity, or unity, that is, an element 1 so that for all a in R,
a*1 = 1*a = a

(Many authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities unitary rings. Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are called associative rings. In this encyclopedia, associativity and the existence of a multiplicative identity are taken to be part of the definition of a ring.)

Note that the commutative law,

a*b=b*a for all a,b in R
is not among the ring axioms listed above; rings that satisfy this law (such as the ring of integers) are called commutative rings. In general, rings are not commutative, though (see, for example, Matrix rings, described below).

The identity element with respect to + is called the zero element of the ring and written as 0. The symbol * is usually omitted from the notation, and the standard order of operation rules are used, so that e.g. a+bc is an abbreviation for a+(b*c). The additive inverse of the element x in a ring is written as -x.

In a ring we have 0=1 if and only if we are dealing with the trivial ring {0} with a single element. Unless specified, all rings in Wikipedia have different 1 and 0.

An element a in a ring is called a unit if it is invertible with respect to multiplication, i.e., if there is an element b in the ring such that

ab = ba = 1
If that is the case, then b is uniquely determined by a and we write a-1 = b.


Simple theorems

From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have

Constructing new rings from given ones

(r1,s1)+(r2,s2) = (r1+r2,s1+s2) and
(r1,s1)(r2,s2) = (r1r2,s1s2).

Glossary and related topics

See Glossary of ring theory for more definitions in ring theory.