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Local ring

A local ring is a ring which has a unique maximal left ideal. In this case the ring also has a unique maximal right ideal, which is a two-sided maximal ideal. In some circles it is often assumed that a local ring should be (left and right) Noetherian, and the non-Noetherian rings are then called `quasi-local'.

The kind of example that motivates this definition is the 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 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 commutative rings.

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