# Local ring > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Local_ring > Markdown URL: https://mediated.wiki/source/Local_ring.md > Source: https://en.wikipedia.org/wiki/Local_ring > Source revision: 1354611255 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|(Mathematical) ring with a unique maximal ideal}} In [mathematics](/source/mathematics), more specifically in [ring theory](/source/ring_theory), '''local rings''' are certain [rings](/source/ring_(mathematics)) that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on [algebraic varieties](/source/algebraic_varieties) or [manifold](/source/manifold)s, or of [algebraic number fields](/source/algebraic_number_fields) examined at a particular [place](/source/place_(mathematics)), or prime. '''Local algebra''' is the branch of [commutative algebra](/source/commutative_algebra) that studies [commutative](/source/commutative_ring) local rings and their [modules](/source/module_(mathematics)). In practice, a commutative local ring often arises as the result of the [localization of a ring](/source/localization_of_a_ring) at a [prime ideal](/source/prime_ideal). The concept of local rings was introduced by [Wolfgang Krull](/source/Wolfgang_Krull) in 1938 under the name ''Stellenringe''. {{cite journal | last = Krull | first = Wolfgang | author-link = Wolfgang Krull | title = Dimensionstheorie in Stellenringen | journal = J. Reine Angew. Math. | volume = 1938 | issue = 179 | page = 204 | year = 1938 | language = de | doi = 10.1515/crll.1938.179.204 | s2cid = 115691729 }} The English term ''local ring'' is due to [Zariski](/source/Zariski). {{cite journal | last = Zariski | first = Oscar | author-link = Oscar Zariski |date=May 1943 | title = Foundations of a General Theory of Birational Correspondences | journal = Trans. Amer. Math. Soc. | volume = 53 | issue = 3 | doi = 10.2307/1990215 | jstor = 1990215 | publisher = American Mathematical Society | pages = 490–542 [497] | url = http://www.ams.org/tran/1943-053-03/S0002-9947-1943-0008468-9/S0002-9947-1943-0008468-9.pdf | doi-access = free }} == Definition and first consequences == A [ring](/source/ring_(mathematics)) ''R'' is a '''local ring''' if it has any one of the following equivalent properties: * ''R'' has a unique [maximal](/source/maximal_ideal) left [ideal](/source/ring_ideal). * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non-[unit](/source/unit_(algebra))s in ''R'' is a non-unit. * 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or {{nowrap|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](/source/Jacobson_radical). The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,Lam (2001), p. 295, Thm. 19.1. 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](/source/coprime) proper ([principal](/source/principal_ideal)) (left) ideals, where two ideals ''I''1, ''I''2 are called ''coprime'' if {{nowrap|1=''R'' = ''I''1 + ''I''2}}. In the case of [commutative ring](/source/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. Before about 1960 many authors required that a local ring be (left and right) [Noetherian](/source/Noetherian_ring), 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](/source/integral_domain) is called a '''local domain'''. == Examples == *All [field](/source/field_(mathematics))s (and [skew field](/source/skew_field)s) are local rings, since {0} is the only maximal ideal in these rings. *The ring \mathbb{Z}/p^n\mathbb{Z} is a local ring ({{mvar|p}} prime, {{math|''n'' ≥ 1}}). The unique maximal ideal consists of all multiples of {{mvar|p}}. *More generally, a nonzero ring in which every element is either a unit or [nilpotent](/source/nilpotent) is a local ring. *An important class of local rings are [discrete valuation ring](/source/discrete_valuation_ring)s, which are local [principal ideal domain](/source/principal_ideal_domain)s that are not fields. *The ring \mathbb{C}[x](/source/x), whose elements are infinite series \sum_{i=0}^\infty a_ix^i where multiplications are given by (\sum_{i=0}^\infty a_ix^i)(\sum_{i=0}^\infty b_ix^i)=\sum_{i=0}^\infty c_ix^i such that c_n=\sum_{i+j=n}a_ib_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](/source/formal_power_series) over a local ring is local; the maximal ideal consists of those power series with [constant term](/source/constant_term) in the maximal ideal of the base ring. *Similarly, the [algebra](/source/algebra_over_a_field) of [dual numbers](/source/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](/source/quotient_ring) ''F''[''X'']/(''X''''n'') 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](/source/geometric_series) to invert all other polynomials [modulo](/source/Ideal_(ring_theory)) ''X''''n''. If ''F'' is a field, then elements of ''F''[''X'']/(''X''''n'') are either [nilpotent](/source/nilpotent) or [invertible](/source/invertible). (The dual numbers over ''F'' correspond to the case {{nowrap|1=''n'' = 2}}.) *Nonzero quotient rings of local rings are local. *The ring of [rational number](/source/rational_number)s with [odd](/source/odd_number) denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is \mathbb{Z}_{(2)}, the integers [localized](/source/localization_of_a_ring) at 2. *More generally, given any [commutative ring](/source/commutative_ring) ''R'' and any [prime ideal](/source/prime_ideal) ''P'' of ''R'', the [localization](/source/localization_of_a_ring) 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 === {{Expand section|date=January 2022}} *The [ring of polynomials](/source/Polynomial_ring) K[x] over a field K is not local, since x and 1 - x are non-units, but their sum is a unit. *The ring of integers \Z is not local since it has a maximal ideal (p) for every prime p. *\Z/(''pq'')\Z, where ''p'' and ''q'' are distinct prime numbers. Both (''p'') and (''q'') are maximal ideals here. === Ring of germs === {{main|Germ (mathematics)}} To motivate the name "local" for these rings, we consider real-valued [continuous function](/source/continuous_function)s defined on some [open interval](/source/interval_(mathematics)) around 0 of the [real line](/source/real_line). We are only interested in the behavior of these functions near 0 (their "local behavior") 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](/source/equivalence_relation), and the [equivalence class](/source/equivalence_class)es are what are called the "[germs](/source/germ_(mathematics)) of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring. To see that this ring of germs is local, we need to characterize its invertible elements. A germ f is invertible if and only if f(0)\neq 0. The reason: if f(0)\neq 0, then by continuity there is an open interval around 0 where f is non-zero, and we can form the function g(x)=\frac 1{f(x)} on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. (Conversely, if f is invertible, then there is some g such that f(0)g(0)=1, hence f(0)\neq 0.) 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. Exactly the same arguments work for the ring of germs of continuous real-valued functions on any [topological space](/source/topological_space) at a given point, or the ring of germs of [differentiable](/source/differentiable) functions on any [differentiable manifold](/source/differentiable_manifold) at a given point, or the ring of germs of [rational function](/source/rational_function)s on any [algebraic variety](/source/algebraic_variety) at a given point. All these rings are therefore local. These examples help to explain why [scheme](/source/scheme_(mathematics))s, the generalizations of varieties, are defined as special [locally ringed space](/source/locally_ringed_space)s. === Valuation theory === {{main|Valuation (algebra)}} Local rings play a major role in valuation theory. By definition, a [valuation ring](/source/valuation_ring) of a field ''K'' is a subring ''R'' such that for every non-zero element ''x'' of ''K'', at least one of ''x'' and ''x''−1 is in ''R''. Any such subring will be a local ring. For example, the ring of [rational number](/source/rational_number)s with [odd](/source/odd_number) denominator (mentioned above) is a valuation ring in \mathbb{Q}. Given a field ''K'', which may or may not be a [function field](/source/Function_field_of_an_algebraic_variety), we may look for local rings in it. If ''K'' were indeed the function field of an [algebraic variety](/source/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 :''F''(''P'') = ''G''(''P'') = 0, the function :''F''/''G'' is an [indeterminate form](/source/indeterminate_form) at ''P''. Considering a simple example, such as :''Y''/''X'', approached along a line :''Y'' = ''tX'', one sees that the ''value at'' ''P'' is a concept without a simple definition. It is replaced by using valuations. === Non-commutative === Non-commutative local rings arise naturally as [endomorphism ring](/source/endomorphism_ring)s in the study of [direct sum](/source/Direct_sum_of_modules) decompositions of [modules](/source/module_(mathematics)) over some other rings. Specifically, if the endomorphism ring of the module ''M'' is local, then ''M'' is [indecomposable](/source/indecomposable_module); conversely, if the module ''M'' has finite [length](/source/length_of_a_module) and is indecomposable, then its endomorphism ring is local. If ''k'' is a [field](/source/field_(mathematics)) of [characteristic](/source/characteristic_(algebra)) {{nowrap|''p'' > 0}} and ''G'' is a finite [''p''-group](/source/p-group), then the [group algebra](/source/group_ring) ''kG'' is local. == Some facts and definitions == === Commutative case=== We also write {{nowrap|(''R'', ''m'')}} for a commutative local ring ''R'' with maximal ideal ''m''. Every such ring becomes a [topological ring](/source/topological_ring) in a natural way if one takes the powers of ''m'' as a [neighborhood base](/source/neighborhood_base) of 0. This is the [''m''-adic topology](/source/I-adic_topology) on ''R''. If {{nowrap|(''R'', ''m'')}} is a commutative [Noetherian](/source/Noetherian_ring) local ring, then :\bigcap_{i=1}^\infty m^i = \{0\} ('''Krull's intersection theorem'''), and it follows that ''R'' with the ''m''-adic topology is a [Hausdorff space](/source/Hausdorff_space). The theorem is a consequence of the [Artin–Rees lemma](/source/Artin%E2%80%93Rees_lemma) together with [Nakayama's lemma](/source/Nakayama's_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 (x). Then a nonzero function e^{-{1 \over x^2}} belongs to m^n for any ''n'', since that function divided by x^n is still smooth. As for any topological ring, one can ask whether {{nowrap|(''R'', ''m'')}} is [complete](/source/Complete_uniform_space) (as a [uniform space](/source/uniform_space)); if it is not, one considers its [completion](/source/Completion_(ring_theory)), again a local ring. Complete Noetherian local rings are classified by the [Cohen structure theorem](/source/Cohen_structure_theorem). In [algebraic geometry](/source/algebraic_geometry), especially when ''R'' is the local ring of a scheme at some point ''P'', {{nowrap|''R'' / ''m''}} is called the ''[residue field](/source/residue_field)'' of the local ring or residue field of the point ''P''. If {{nowrap|(''R'', ''m'')}} and {{nowrap|(''S'', ''n'')}} are local rings, then a '''local ring homomorphism''' from ''R'' to ''S'' is a [ring homomorphism](/source/ring_homomorphism) {{nowrap|''f'' : ''R'' → ''S''}} with the property {{nowrap|''f''(''m'') ⊆ ''n''}}.{{Cite web|url=http://stacks.math.columbia.edu/tag/07BI|title=Tag 07BI}} These are precisely the ring homomorphisms that are continuous with respect to the given topologies on ''R'' and ''S''. For example, consider the ring morphism \mathbb{C}[x]/(x^3) \to \mathbb{C}[x,y]/(x^3,x^2y,y^4) sending x \mapsto x. The preimage of (x,y) is (x). Another example of a local ring morphism is given by \mathbb{C}[x]/(x^3) \to \mathbb{C}[x]/(x^2). === General case=== The [Jacobson radical](/source/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.The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals. For an element ''x'' of the local ring ''R'', the following are equivalent: * ''x'' has a left inverse * ''x'' has a right inverse * ''x'' is invertible * ''x'' is not in ''m''. If {{nowrap|(''R'', ''m'')}} is local, then the [factor ring](/source/factor_ring) ''R''/''m'' is a [skew field](/source/skew_field). If {{nowrap|''J'' ≠ ''R''}} is any two-sided ideal in ''R'', then the factor ring ''R''/''J'' is again local, with maximal ideal ''m''/''J''. A [deep theorem](/source/Kaplansky's_theorem_on_projective_modules) by [Irving Kaplansky](/source/Irving_Kaplansky) says that any [projective module](/source/projective_module) over a local ring is [free](/source/free_module), though the case where the module is finitely-generated is a simple corollary to [Nakayama's lemma](/source/Nakayama's_lemma). This has an interesting consequence in terms of [Morita equivalence](/source/Morita_equivalence). Namely, if ''P'' is a [finitely generated](/source/finitely_generated_module) projective ''R'' module, then ''P'' is isomorphic to the free module ''R''''n'', and hence the ring of endomorphisms \mathrm{End}_R(P) is isomorphic to the full ring of matrices \mathrm{M}_n(R). Since every ring Morita equivalent to the local ring ''R'' is of the form \mathrm{End}_R(P) 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''. ==See also== * [Discrete valuation ring](/source/Discrete_valuation_ring) * [Heyting field](/source/Heyting_field) * [Semi-local ring](/source/Semi-local_ring) * [Gorenstein local ring](/source/Gorenstein_local_ring) * [Regular local ring](/source/Regular_local_ring) ==Notes== == References == * {{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}} * {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | publisher=Dover| isbn = 978-0-486-47187-7}} == External links == *[https://mathoverflow.net/q/255511 The philosophy behind local rings] Category:Ring theory Category:Localization (mathematics) --- Adapted from the Wikipedia article [Local ring](https://en.wikipedia.org/wiki/Local_ring) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Local_ring?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.