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| In [[abstract algebra]], more specifically [[ring theory]], '''local rings''' are certain [[ring (mathematics)|rings]] that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on [[algebraic variety|varieties]] or [[manifold]]s, or of [[algebraic number fields]] examined at a particular [[place (mathematics)|place]], or prime. '''Local algebra''' is the branch of [[commutative algebra]] that studies local rings and their [[module (mathematics)|modules]].
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| In practice, a commutative local ring often arises as the result of the [[localization of a ring]] at a prime ideal.
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| The concept of local rings was introduced by [[Wolfgang Krull]] in 1938 under the name ''Stellenringe''.<ref name="Krull">
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| {{cite journal
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| | last = Krull
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| | first = Wolfgang
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| | authorlink = Wolfgang Krull
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| | title = Dimensionstheorie in Stellenringen
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| | journal = J. Reine Angew. Math.
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| | volume = 179
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| | page = 204
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| | year = 1938
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| | language = German
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| }}</ref> The English term ''local ring'' is due to [[Zariski]].<ref name = "Zariski">
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| {{cite journal
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| | last = Zariski
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| | first = Oscar
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| | authorlink = Oscar Zariski
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| |date=May 1943
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| | title = Foundations of a General Theory of Birational Correspondences
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| | journal = Trans. Amer. Math. Soc.
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| | volume = 53
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| | issue = 3
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| | doi = 10.2307/1990215
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| | jstor = 1990215
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| | publisher = American Mathematical Society
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| | pages = 490–542 [497]
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| }}</ref>
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| == Definition and first consequences ==
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| A [[ring (mathematics)|ring]] ''R'' is a '''local ring''' if it has any one of the following equivalent properties:
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| * ''R'' has a unique maximal [[ring ideal|left ideal]].
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| * ''R'' has a unique [[maximal ideal|maximal right ideal]].
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| * 1 ≠ 0 and the sum of any two non-[[unit (algebra)|unit]]s in ''R'' is a non-unit.
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| * 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or 1 − ''x'' is a unit.
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| * 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).
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| 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,<ref>Lam (2001), p. 295, Thm. 19.1.</ref> 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 ideal|principal]]) (left) ideals where two ideals ''I''<sub>1</sub>, ''I''<sub>2</sub> are called ''coprime'' if ''R'' = ''I''<sub>1</sub> + ''I''<sub>2</sub>.
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| In the case of [[commutative ring]]s, 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.
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| Before about 1960 many authors required that a local ring be (left and right) [[Noetherian ring|Noetherian]], and (possibly non-Noetherian) local rings were called '''quasi-local rings'''. In this article this requirement is not imposed.
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| A local ring that is an [[integral domain]] is called a '''local domain'''.
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| == Examples ==
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| *All [[field (mathematics)|field]]s (and [[skew field]]s) are local rings, since {0} is the only maximal ideal in these rings.
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| *A nonzero ring in which every element is either a unit or nilpotent is a local ring.
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| *An important class of local rings are [[discrete valuation ring]]s, which are local [[principal ideal domain]]s that are not fields.
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| *Every ring of [[formal power series]] over a field (even in several variables) is local; the maximal ideal consists of those power series without [[constant term]].
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| *Similarly, the algebra of [[dual numbers]] over any field is local. More generally, if ''F'' is a field and ''n'' is a positive integer, then the [[quotient ring]] ''F''[''X'']/(''X''<sup>''n''</sup>) is local with maximal ideal consisting of the classes of polynomials with zero constant term, since one can use a [[geometric series]] to invert all other polynomials [[Ideal (ring theory)|modulo]] ''X''<sup>''n''</sup>. In these cases elements are either [[nilpotent]] or [[invertible]]. (The dual numbers over ''F'' is the case ''n''=2.)
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| *The ring of [[rational number]]s with [[odd number|odd]] denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator: this is the integers [[localization of a ring|localized]] at 2.
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| More generally, given any [[commutative ring]] ''R'' and any [[prime ideal]] ''P'' of ''R'', the [[localization of a ring|localization]] of ''R'' at ''P'' is local; the maximal ideal is the ideal generated by ''P'' in this localization.
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| === Ring of germs ===
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| {{main|Germ (mathematics)}}
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| To motivate the name "local" for these rings, we consider real-valued [[continuous function]]s defined on some [[interval (mathematics)|open interval]] around 0 of the [[real line]]. We are only interested in the local behavior of these functions near 0 and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an [[equivalence relation]], and the [[equivalence class]]es are the "[[germ (mathematics)|germs]] of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.
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| To see that this ring of germs is local, we need to identify its invertible elements. A germ ''f'' is invertible if and only if ''f''(0) ≠ 0. The reason: if ''f''(0) ≠ 0, then there is an open interval around 0 where ''f'' is non-zero, and we can form the function ''g''(''x'') = 1/''f''(''x'') on this interval. The function ''g'' gives rise to a germ, and the product of ''fg'' is equal to 1.
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| With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs ''f'' with ''f''(0) = 0.
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| Exactly the same arguments work for the ring of germs of continuous real-valued functions on any [[topological space]] at a given point, or the ring of germs of differentiable functions on any differentiable [[manifold]] at a given point, or the ring of germs of rational functions on any [[algebraic variety]] at a given point. All these rings are therefore local. These examples help to explain why [[scheme (mathematics)|scheme]]s, the generalizations of varieties, are defined as special [[locally ringed space]]s.
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| === Valuation theory ===
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| {{main|Valuation (algebra)}}
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| Local rings play a major role in valuation theory. By definition, a [[valuation ring]] of ''K'' is a subring ''R'' such that for every non-zero element ''x'' of ''K'', at least one of ''x'' and ''x''<sup>−1</sup> is in ''R''. Any such subring will be a local ring. For example, the ring of [[rational number]]s with [[odd number|odd]] denominator (mentioned above) is a valuation ring in <math>\mathbb{Q}</math>.
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| Given a field ''K'', which may or may not be a [[Function field of an algebraic variety|function field]], we may look for local rings in it. If ''K'' were indeed the function field of an [[algebraic variety]] ''V'', then for each point ''P'' of ''V'' we could try to define a valuation ring ''R'' of functions "defined at" ''P''. In cases where ''V'' has dimension 2 or more there is a difficulty that is seen this way: if ''F'' and ''G'' are rational functions on ''V'' with
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| :''F''(''P'') = ''G''(''P'') = 0,
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| the function
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| :''F''/''G''
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| is an [[indeterminate form]] at ''P''. Considering a simple example, such as
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| :''Y''/''X'',
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| approached along a line
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| :''Y'' = ''tX'',
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| one sees that the ''value at'' ''P'' is a concept without a simple definition. It is replaced by using valuations.
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| === Non-commutative ===
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| Non-commutative local rings arise naturally as [[endomorphism ring]]s in the study of [[Direct sum of modules|direct sum]] decompositions of [[module (mathematics)|modules]] over some other rings. Specifically, if the endomorphism ring of the module ''M'' is local, then ''M'' is [[indecomposable module|indecomposable]]; conversely, if the module ''M'' has finite [[length of a module|length]] and is indecomposable, then its endomorphism ring is local.
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| If ''k'' is a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] ''p'' > 0 and ''G'' is a finite [[p-group|''p''-group]], then the [[group algebra]] ''kG'' is local.
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| == Some facts and definitions ==
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| === Commutative Case===
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| We also write (''R'', ''m'') for a commutative local ring ''R'' with maximal ideal ''m''. Every such ring becomes a [[topological ring]] in a natural way if one takes the powers of ''m'' as a [[neighborhood base]] of 0. This is the [[I-adic topology|''m''-adic topology]] on ''R''.
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| If (''R'', ''m'') and (''S'', ''n'') are local rings, then a '''local ring homomorphism''' from ''R'' to ''S'' is a [[ring homomorphism]] ''f'' : ''R'' → ''S'' with the property ''f''(''m'') ⊆ ''n''. These are precisely the ring homomorphisms which are continuous with respect to the given topologies on ''R'' and ''S''.
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| A ring homomorphism ''f'' : ''R'' → ''S'' is a local ring homomorphism if and only if <math>f^{-1}(n)=m</math>; that is, the preimage of the maximal ideal is maximal.
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| As for any topological ring, one can ask whether (''R'', ''m'') is [[completeness (topology)|complete]] (as a topological space); if it is not, one considers its [[Completion (ring theory)|completion]], again a local ring.
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| If (''R'', ''m'') is a commutative [[Noetherian ring|Noetherian]] local ring, then
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| :<math>\bigcap_{i=1}^\infty m^i = \{0\}</math> | |
| ('''Krull's intersection theorem'''), and it follows that ''R'' with the ''m''-adic topology is a [[Hausdorff space]]. The theorem is a consequence of the [[Artin–Rees lemma]], and, as such, the "Noetherian" assumption is crucial. Indeed, let ''R'' be the ring of germs of infinitely differentiable functions at 0 in the real line and ''m'' be the maximal ideal <math>(x)</math>. Then a nonzero function <math>e^{-{1 \over x^2}}</math> belongs to <math>m^n</math> for any ''n'', since that function divided by <math>x^n</math> is still smooth.
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| In algebraic geometry, especially when ''R'' is the local ring of a scheme at some point ''P'', ''R / m'' is called the ''[[residue field]]'' of the local ring or residue field of the point ''P''.
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| === General Case===
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| The [[Jacobson radical]] ''m'' of a local ring ''R'' (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of ''R''. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.<ref>The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.</ref>
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| For an element ''x'' of the local ring ''R'', the following are equivalent:
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| * ''x'' has a left inverse
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| * ''x'' has a right inverse
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| * ''x'' is invertible
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| * ''x'' is not in ''m''.
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| If (''R'', ''m'') is local, then the [[factor ring]] ''R''/''m'' is a [[skew field]]. If ''J'' ≠ ''R'' is any two-sided ideal in ''R'', then the factor ring ''R''/''J'' is again local, with maximal ideal ''m''/''J''.
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| A deep theorem by [[Irving Kaplansky]] says that any [[projective module]] over a local ring is [[free module|free]], though the case where the module is finitely-generated is a simple corollary to [[Nakayama's lemma]]. This has an interesting consequence in terms of [[Morita equivalence]]. Namely, if ''P'' is a [[finitely generated module|finitely generated]] projective ''R'' module, then ''P'' is isomorphic to the free module ''R''<sup>''n''</sup>, and hence the ring of endomorphisms <math>\mathrm{End}_R(P)</math> is isomorphic to the full ring of matrices <math>\mathrm{M}_n(R)</math>. Since every ring Morita equivalent to the local ring ''R'' is of the form <math>\mathrm{End}_R(P)</math> for such a ''P'', the conclusion is that the only rings Morita equivalent to a local ring ''R'' are (isomorphic to) the matrix rings over ''R''.
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| ==Notes==
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| <references/>
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| == References ==
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| * {{Cite book| last=Lam| first=T.Y.| author-link=T.Y. Lam| year=2001| title= A first course in noncommutative rings| edition=2nd| series= Graduate Texts in Mathematics| publisher=Springer-Verlag| isbn = 0-387-95183-0}}
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| * {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7}}
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| ==See also==
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| *[[Discrete valuation ring]]
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| *[[Semi-local ring]]
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| *[[Valuation ring]]
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| [[Category:Ring theory]]
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| [[Category:Localization (mathematics)]]
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