Characteristic of a ring – Wikipedia

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

${displaystyle { n.1_{A}~=~underbrace {1_{A}+1_{A}+cdots +1_{A}} _{n;{text{termes}}}~=~0_{A}}}$

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 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 ,

${DisplayStyle Underbrace {A +CDOTS +A} _ {n {text {times}} = 0.}$

Yes a tel n does not exist, the characteristic is 0.

There is a unique unit rings morphism

${displaystyle f}$

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:

${displaystyle f(n)=1_{A}+cdots +1_{A}}$

,

where 1 _A is repeated n time.
As ℤ is a Euclidean ring, the nucleus of

${displaystyle f}$

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

${displaystyle f}$

either ideal n ℤ.

Indeed, if

${displaystyle mathbb {z} /nmathbb {z}}$

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 → B , the characteristic of B divides that of A .
  Indeed, the homomorphism of unit rings
  ${displaystyle mathbb {Z} to B}$

  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

  ${Displastyle qmathbb {z}}$

  , or g ( f ( p )) = g (0 _A ) = 0 _B , although

  ${Displastyle qmathbb {z}}$

  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 × 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

${displaystyle mathbb {z} /pmmbb {z}}$

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
  ${displaystyle mathbb {Q} }$

  .

  Indeed, such a body K already contains (like any zero characteristic ring) a copy of

  ${displaystyle mathbb {z}}$

  . As K is a body, it therefore contains the body of the fractions of

  ${displaystyle mathbb {z}}$

  , namely the body

  ${displaystyle mathbb {Q} }$

  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

  ${displaystyle mathbb {Q} }$

  .

• 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 .