[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/characteristic-of-a-ring-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/characteristic-of-a-ring-wikipedia\/","headline":"Characteristic of a ring – Wikipedia","name":"Characteristic of a ring – Wikipedia","description":"before-content-x4 A wikipedia article, free l’encyclop\u00e9i. after-content-x4 In algebra, the characteristic a ring (unitary) A is by definition the order","datePublished":"2019-10-28","dateModified":"2019-10-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/089ed508b71bdd21f7ee175b6de693774cd80887","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/089ed508b71bdd21f7ee175b6de693774cd80887","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/characteristic-of-a-ring-wikipedia\/","wordCount":3697,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4A wikipedia article, free l’encyclop\u00e9i.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In algebra, the characteristic a ring (unitary) A is by definition the order for the additive law of the neutral element of the multiplicative law if this order is finished; If this order is infinite, the characteristic of the ring is by definition zero. We note, for a unitary ring ( A , +, \u00d7), 0 A the neutral element of “+” and 1 A to the “\u00d7”.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The characteristic of a ring A is therefore the smallest integer n > 0 such that \u00a0n.1A\u00a0=\u00a01A+1A+\u22ef+1A\u23dfntermes\u00a0=\u00a00A{displaystyle { n.1_{A}~=~underbrace {1_{A}+1_{A}+cdots +1_{A}} _{n;{text{termes}}}~=~0_{A}}} If such an integer exists. Otherwise (in other words if 1 A is infinite), the characteristic is zero. The subanary of A engendered by 1 A , called the First subanary [ first ] of A , is isomorphic to \u2124\/ c \u2124, where c is the characteristic of A .        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4When the ring A is honest and of non-zero characteristic, this characteristic is a primary number and this first subanary is a finished body, called the First sub-body of A . Note 1 : This definition complies with works published in XXI It is century [ 2 ] . Bourbaki [ 3 ] Explicitly says only define the characteristic of a ring if this ring contains a body. Lang [ 4 ] consider the ideal of \u2124 formed by n such as n .first A = 0; If this ideal is first, that is to say the form c \u2124 Where c is zero or a primary number, it defines the characteristic of A as being the number c . He does not define it otherwise. Note 2 : Some authors do not require the presence of a unitary element in the definition of a ring (see the detailed article), a structure often called pseudo-year . In this case, the previous definition must be replaced by the following, more general. The characteristic of A is the smallest n , if exists, such as, for any element a of A , a+\u22ef+a\u23dfn\u00a0fois= 0. {DisplayStyle Underbrace {A +CDOTS +A} _ {n {text {times}} = 0.} Yes a tel n does not exist, the characteristic is 0.   There is a unique unit rings morphism f {displaystyle f} of \u2124 in A (\u2124 is indeed an initial object of the category of the rings). By definition, if n is a strictly positive integer, we have:  f ( n ) = first A+ \u22ef + first A{displaystyle f(n)=1_{A}+cdots +1_{A}} ,  where 1 A is repeated n time.As \u2124 is a Euclidean ring, the nucleus of f {displaystyle f} is a main ideal and, by definition, characteristic of A is its positive generator. More explicitly, it is the only natural whole n such as the core of f {displaystyle f} either ideal n \u2124. Indeed, if Z\/n Z{displaystyle mathbb {z} \/nmathbb {z}} is a unit subanary of a ring integrated then it is itself an integrated, therefore n is zero or first. For any morphism of unit rings g : A \u2192 B , the characteristic of B divides that of A . Indeed, the homomorphism of unit rings Z\u2192 B {displaystyle mathbb {Z} to B} is the compound homomorphism g \u2218 f . And p And q are the respective characteristics of A and of B , the nucleus of g \u2218 f is therefore q Z{Displastyle qmathbb {z}} , or g ( f ( p )) = g (0 A ) = 0 B , although q Z{Displastyle qmathbb {z}} contains p , in other words q uniform p . And A is a ring commutatif , and if its characteristic is a primary number p , then for all elements x, y In A , on a ( x + y ) p = x p + and p . The application that x associated x p is an endomorphism of ring called Endomorphism of Frobenius. The result immediately stems from the Newton’s pair formula and what p Divide the binomial coefficients appearing in development. The characteristic of a rings product A \u00d7 B is the P.P.C.M of the characteristics of these rings. As with any integrated ring, the characteristic of a body K is either 0 or a primary number p . In addition, in the second case, as for any characteristic ring p not nulle, K contains a copy of WITH \/ p WITH {displaystyle mathbb {z} \/pmmbb {z}} who (since here p is first) is a body: it is the only finished body F p To p elements. Any zero characteristic body contains a copy of Q{displaystyle mathbb {Q} } .  Indeed, such a body K already contains (like any zero characteristic ring) a copy of Z{displaystyle mathbb {z}} . As K is a body, it therefore contains the body of the fractions of Z{displaystyle mathbb {z}} , namely the body Q{displaystyle mathbb {Q} } rational. Any body therefore has a minimal sub-body, its first body , isomorphic (according to its characteristic) to a finished body F p or to the body Q{displaystyle mathbb {Q} } . Any finished body is characteristic of a primary number, and for cardinal a power of this number. And K is a finished body it is, like any finished ring, of non -zero characteristic. By the above, its characteristic is therefore a primary number p And K Contains a copy of the body F p . Fact, K is a vector space on F p . So his cardinal is p to power its dimension (which, therefore, is necessarily over, in other words K is a finite extension of F p ). For any first number p , there are infinite bodies of characteristic p  :for example the body of rational fractions on F p or the algebraic fence of F p . \u2191 (in) Shreeram Shankar Abhyankar, Lectures on Algebra , World Scientific, 2006 ( read online ) , p. 21 .  \u2191 For example (in) Joseph Gallian (in) , Contemporary Abstract Albegra , Cengage Learning, 2010 , 656 p.  (ISBN\u00a0 978-0-547-16509-7 , read online ) , p. 252-253 .  \u2191 N. Bourbaki, Algebra, chapters 4 to 7 , Masson, 1981 , V.2 .  \u2191 Serge Lang , Algebra  [Detail of editions] , 2004, p. 97 .         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