In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. The English term local ring is due to Zariski.
Definition and first consequences A ring R is a local ring if it has any one of the following equivalent properties:
R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non-units in R is a non-unit. 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit. If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0). If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if R = I1 + I2. In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal. Before about 1960 many authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed. A local ring that is an integral domain is called a local domain.
Examples All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. The ring Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } is a local ring (p prime, n ≥ 1). The unique maximal ideal consists of all multiples of p. More generally, a nonzero ring in which every element is either a unit or nilpotent is a local ring. An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields. The ring C [ [ x ] ] {\displaystyle \mathbb {C} [[x]]} , whose elements are infinite series ∑ i = 0 ∞ a i x i {\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}} where multiplications are given by ( ∑ i = 0 ∞ a i x i ) ( ∑ i = 0 ∞ b i x i ) = ∑ i = 0 ∞ c i x i {\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}} such that c n = ∑ i + j = n a i b j {\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}} , is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero. More generally, every ring of formal power series over a local ring is local; the maximal ideal consists of those power series with constant term in the maximal ideal of the base ring. Similarly, the algebra of dual numbers over any field is local. More generally, if F is a local ring and n is a positive integer, then the quotient ring F[X]/(Xn) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a geometric series to invert all other polynomials modulo Xn. If F is a field, then elements of F[X]/(Xn) are either nilpotent or invertible. (The dual numbers over F correspond to the case n = 2.) Nonzero quotient rings of local rings are local. The ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is Z ( 2 ) {\displaystyle \mathbb {Z} _{(2)}} , the integers localized at 2. More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization; that is, the maximal ideal consists of all elements a/s with a ∈ P and s ∈ R - P.
Non-examples
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