ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# Local ring

In abstract algebra, a local ring is a ring which has a unique maximal left ideal.

Some authors require that a local ring be (left and right) Noetherian, and the non-Noetherian rings are then called `quasi-local'. Wikipedia does not impose this requirement.

## Properties

Every local ring also has a unique maximal right ideal, and this right ideal is equal to the unique maximal left ideal (and equal to the ring's Jacobson radical) and hence a two-sided maximal ideal. (In the non-commutative case, this two-sided maximal ideal need not be the only one, however.)

The situation for commutative rings is simpler: a commutative ring is local if and only if it has a unique maximal ideal. This maximal ideal contains then precisely the non-units of the ring. In fact, a commutative ring is local if and only if the sum of two non-units is always again a non-unit.

## Examples

All fields and skew fields are local rings, since {0} is the only maximal ideal in these rings.

The kind of example that motivates the definition is the commutative ring R of real-valued continuous functions defined on some interval around 0 of the real line. The idea is that R will have a maximal ideal m consisting of all functions f in R with f(0) = 0. That m really is a maximal ideal follows easily from identifying the factor ring R/m with the field of real numbers.

To understand why R should just have this one maximal ideal, we translate that into the statement that any f in R outside m should be invertible, i.e. have a multiplicative inverse in R. This we can prove, by paying close attention to the characterisation of functions in R.

So assume f(0) is not 0 and define g by g(x) = 1/f(x) on some small interval around 0: this is a proper definition since f is continuous. We want to say that fg = 1. In fact it is 1 wherever it is defined. We have to understand that 1, the multiplicative identity in R. means a function taking the constant value 1 on some unspecified interval around 0. In order for that to work we must have 1.f = f, and that entails only considering the values of f near 0. Therefore we should identify two functions if they coincide on any interval containing 0. Then we do have a natural example of a local ring, which consists of functions (strictly, germs of functions) considered only in terms of their local behaviour at one point.

If we restricted to polynomials in R the definition would be easier, since two polynomials coinciding on a whole interval are identical. But to have the multiplicative inverses, we should make that rational functions. In that way we get the kind of example used in algebraic geometry.

Other examples of commutative local rings include the ring of rational numbers with odd denominator, and more generally the localization of any commutative ring at a prime ideal.

Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings.

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.