The Commutative ring reference article from the English Wikipedia on 24-Jul-2004
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Commutative ring

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In Ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as *, then a*b=b*a.

The study of commutative rings is called commutative algebra.

Table of contents
1 History
2 Examples
3 Constructing new commutative rings from given ones
4 General Discussion

History

See Ring theory

Examples

is not equal to the multiplication performed in the opposite order:

Constructing new commutative rings from given ones

General Discussion

The inner structure of a commutative ring is determined by considering its ideals. All ideals in a commutative ring are two-sided, which makes considerations considerably easier than in the general case.

The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules. This subject is significantly more difficult when the commutative ring is not a field and is usually called homological algebra. The set of ideals within a commutative ring R can be exactly characterized as the set of R-modules which are subsets of R.

Commutative rings are sometimes characterized by the elements they contain which have special properties. A multiplicative identity in a commutative ring is a special element (usually denoted 1) having the property that for every element a of the ring, 1*a = a. A commutative ring possessing such an element is said to be a ring with identity.

An element a of a commutative ring (with identity) is called a unit if it possesses a multiplicative inverse, i.e., if there exists another element b of the ring (with b not necessarily distinct from a) so that a*b = b*a = 1. Every nonzero element of a field is a unit. Every element of a commutative local ring not contained in the maximal ideal is a unit.

An non-zero element a of a commutative ring is said to be a zero divisor if there exists another non-zero element b of the ring (b not necessarily distinct from a) so that a*b = 0. A commutative ring with identity which possesses no zero divisors is called an integral domain since it closely resembles the integers in some ways.