Characteristic of a ring – Wikipedia
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In 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 , +, ×), 0 A the neutral element of “+” and 1 A to the “×”.
The characteristic of a ring A is therefore the smallest integer n > 0 such that
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 ℤ/ c ℤ, where c is the characteristic of A .
When 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 ℤ formed by n such as n .first A = 0; If this ideal is first, that is to say the form c ℤ 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 ,
Yes a tel n does not exist, the characteristic is 0.
There is a unique unit rings morphism
of ℤ in A (ℤ is indeed an initial object of the category of the rings). By definition, if n is a strictly positive integer, we have:
where 1 A is repeated n time.
As ℤ is a Euclidean ring, the nucleus of
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
either ideal n ℤ.
- Indeed, if 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 → B , the characteristic of B divides that of A .
Indeed, the homomorphism of unit rings is the compound homomorphism g ∘ f . And p And q are the respective characteristics of A and of B , the nucleus of g ∘ f is therefore , or g ( f ( p )) = g (0 A ) = 0 B , although 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 × 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
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 .
- Indeed, such a body K already contains (like any zero characteristic ring) a copy of . As K is a body, it therefore contains the body of the fractions of , namely the body 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 .
- 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 .
- (in) Shreeram Shankar Abhyankar, Lectures on Algebra , World Scientific, ( read online ) , p. 21 .
- For example (in) Joseph Gallian (in) , Contemporary Abstract Albegra , Cengage Learning, , 656 p. (ISBN 978-0-547-16509-7 , read online ) , p. 252-253 .
- N. Bourbaki, Algebra, chapters 4 to 7 , Masson, , V.2 .
- Serge Lang , Algebra [Detail of editions] , 2004, p. 97 .
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