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Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative. A ring is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called addition and multiplication and denoted by "+" and "⋅". To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; the identity elements for addition and multiplication are denoted 1, respectively. If the multiplication is commutative, i.e. a ⋅ b = b ⋅ a,then the ring R is called commutative. In the remainder of this article, all rings will be commutative. An important example, in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication.

As the multiplication of integers is a commutative operation, this is a commutative ring. It is denoted Z as an abbreviation of the German word Zahlen. A field is a commutative ring where every non-zero element a is invertible. Therefore, by definition, any field is a commutative ring; the rational and complex numbers form fields. If R is a given commutative ring the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R; the same holds true for several variables. If V is some topological space, for example a subset of some Rn, real- or complex-valued continuous functions on V form a commutative ring; the same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold. In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element a of ring R is called a unit. Another particular type of element is the zero divisors, i.e. an element a such that there exists a non-zero element b of the ring such that ab = 0.

If R possesses no non-zero zero divisors, it is called an integral domain. An element a satisfying an = 0 for some positive integer n is called nilpotent; the localization of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if S is a multiplicatively closed subset of R the localization of R at S, or ring of fractions with denominators in S denoted S−1R consists of symbols r s with r ∈ R, s ∈ Ssubject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language Q is the localization of Z at all nonzero integers; this construction works for any integral domain R instead of Z. The localization −1R is a field, called the quotient field of R. Many of the following notions exist for not commutative rings, but the definitions and properties are more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably.

For a ring R, an R-module M is like. That is, elements in a module can be added; the study of modules is more involved than the one of vector spaces in linear algebra, since several features of vector spaces fail for modules in general: modules need not be free, i.e. of the form M = ⨁ i ∈ I R. For free modules, the rank of a free module may not be well-defined. Submodules of finitely generated modules need not be finitely generated. Ideals of a ring R are the submodules of R, i.e. the modules contained in R. In more detail, an ideal I is a non-empty subset of R such that for all r in R, i and j in I, both ri and i + j are in I. For various applications, understanding the ideals of a ring is of particular importance, but one proceeds by studying modules in general. Any ring has two ideals, namely the whole ring; these two ideals are the only ones if R is a field. Given any subset F = j ∈ J of R, the ideal generated by F is the smallest ideal that contains F. Equivalently, it is given by finite linear combinations r1f1 + r2f2 +... + rnfn.

If F consists of a single element r, the ideal generated by F consists of the multiples of r, i.e. the elements of the form rs for arbitrary elements s. Such an ideal is called a principal ideal. If every ideal is a principal ideal, R is called a principal ideal ring; these two are in addition domains, so they are called principal ideal domains. Unlike for general rings, for a principal ideal domain, the properties of individual elements are tied to the properties of the ring as a whole. For example, any principal ideal domain R is a unique factorization domain which means that any element is a product of irreducible elements, in a unique way. Here, an element a in a domain

Premios Nuestra Tierra

The Premios Nuestra Tierra are a recognition, given to Colombian artists. They have a format restricted to the Colombian scope; the awards began in 2007 and were created by Blanca Luz Holguín, Fernán Martínez and Alejandro Villalobos with the purpose of encouraging Colombian artists and trying to encourage new Colombian singers so that the music industry in Colombia has much greater movement. In 2008 and 2009 Movistar bought half of the proposal, renamed "Premios Movistar nuestra tierra" in order to promote its other colombian project "Movistar Radio". Since 2010 the prizes have been renamed "premios nuestra tierra" without any sponsored; the winners are chosen through an association that has the organization of the awards, where directors participate, some representatives of the Colombian record companies and executives of RCN and Caracol Radio. The prize is a heart with musical notes around and a crown at the top representing the heart of Jesus, at the beginning it was a plaque that symbolized the signs that Colombian public buses put to inform people where they are going, they realized that the plaque was uncomfortable, so they tried to modernize the logo to something more comfortable like is the prize, awarded General Best Song of the Year Best Artist of the Year.

Best Album of the Year. Best New Artist. Best Producer. Tropical Pop Best Tropical Pop Interpretation. Best Tropical Pop Artist of the Year. Pop Best Pop Artist of the Year Best Pop SongUrban Best Urban Song of the Year Best Urban ArtistAlternative Best Alternative Artist Better Alternative InterpretationVallenato Best Performance Best ArtistTropical Best Solo Artist or Tropical Group Best Tropical InterpretationFolkloric Best Solo Artist or Folkloric Group of the Year Best PerformanceVideos Best Video for Artist Best Music for TV Best Movie Soundtrack:Popular Best Artist of the year Best Song of the Year:Christian Best Christian Song Best Christian Artist:Públic Best Interpretation of the Public Best Public Artist Best Fan Club:Digital Best Website: Twitterer of the Year Best DJ: The awards have been given annually and uninterrupted since 2007, between the months of March or April Latin American television awards Web official

The Nutcracker Suite (Duke Ellington album)

The Nutcracker Suite is an album by American pianist and bandleader Duke Ellington recorded for the Columbia label in 1960 featuring jazz interpretations of "The Nutcracker" by Tchaikovsky, arranged by Ellington and Billy Strayhorn. All compositions by Pyotr Ilyich Tchaikovsky "Overture" - 3:22 "Toot Toot Tootie Toot" - 2:30 "Peanut Brittle Brigade" - 4:37 "Sugar Rum Cherry" - 3:05 "Entr'acte" - 1:53 "Volga Vouty" - 2:52 "Chinoiserie" - 2:50 "Danse of the Floreadores" - 4:04 "Arabesque Cookie" - 5:44Recorded on May 26, May 31, June 3, 21 and 22, 1960. Duke Ellington – piano Willie Cook, Fats Ford, Ray Nance, Clark Terry - trumpet Lawrence Brown, Booty Wood, Britt Woodman - trombone Juan Tizol - valve trombone Jimmy Hamilton - clarinet, tenor saxophone Johnny Hodges - alto saxophone Russell Procope - alto saxophone, clarinet Paul Gonsalves - tenor saxophone Harry Carney - baritone saxophone, bass clarinet Aaron Bell - bass Sam Woodyard - drums