Characteristic of a ring

See also: Characteristic

In Algebra, the characteristic of a unit ring $A$ is by definition the order for the additive law of the neutral element of the multiplicative law. In an abstract way, the characteristic of a ring is the number of times that it is necessary to add the neutral element of the multiplicative law to obtain the neutral element of the additive law.

For a unit ring $(has,\; +,\; \backslash \; times)$, one notes $0\_A$ the neutral element of “$+$” and $1\_A$ that of “$\backslash \; times$”. There exists a single homomorphism of unit rings $f$ of $\backslash \; mathbb\; \{Z\}$ in $A$ ($\backslash \; mathbb\; \{Z\}$ is indeed an initial object of the category of the rings). By definition, if $n$ is a Integer strictly positive, one a:

$f\; (N)\; =1\_A+\; \backslash \; cdots+1\_A\; \backslash ,$, where $1\_A$ is repeated $n$ time. As $\backslash \; mathbb\; \{Z\}$ is a Euclidean Anneau, the core of $f$ is principal and by definition, the characteristic of $A$ is its positive generator. More explicitly, it is single the Integer positive or null $c$ such as the core of $f$ is the ideal $c\; \backslash \; mathbb\; \{Z\}$.

Properties on the rings

• the characteristic of a Anneau integrates is either null, or a Prime number.

Indeed, if the characteristic of a unit ring $A$ is a divisible entirety not no one $p>0$, it can be written: $p\; =\; N\; \backslash \; times\; m$ where $n$ and $m$ are strictly higher than 1. By taking again the notations above, as $f$ is a homomorphism of rings, $f\; (N\; \backslash \; times\; m)\; =f\; (N)\; \backslash \; times\; F\; (m)\; =0$. If $A$ is just, one of the factors $f\; (N)$ or $f\; (m)$ is null. That contradicted the definition of $p$, which, as generating positive of the core of $f$ is the smallest positive entirety cancelled by $f$. Thus $p$ is not divisible, it is first.

• If the characteristic of a ring is null, this one is infinite, because it contains an isomorphous subring with $\backslash \; mathbb\; Z$.

Homomorphism F is injective. It induces an isomorphism on its image which is a unit subring.

• If $B$ is a unit subring of $A$, then $A$ and $B$ have even characteristic.

Homomorphism $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; A$ is factorized obviously through inclusion $A\; \backslash \; rightarrow\; B$.

• For any homomorphism of unit rings $g:\; With\; \backslash \; rightarrow\; B$, the characteristic of $B$ divides that of $A$.

Indeed, the made up one of homomorphisms g.f is the single homomorphism of unit rings $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; B$. Its core is the reciprocal image by F of the core of G . It contains in particular the core of F . If p and Q is the characteristics of has and of B , Q Z contains p Z . Therefore, Q divides p .

• If $A$ is a commutative ring , and if its characteristic is a prime number $p$, then for all elements $x,\; y$ in $A$, one has $(x+y)\; ^p\; =\; x^p+y^p$. The application defined by $f\; (X)\; =x^p$ is a Endomorphisme of injective ring called Endomorphisme de Frobenius.

The result rises immediately from the formula of Newton and what p divided in Z binomial coefficients appearing in the development.

Properties on the bodies

• If a body K is of null characteristic, it contains a copy of Q. If it is of characteristic p, it contains a copy of Z/pZ

As for any just ring, the characteristic of K is either null or a prime number p . In the first case, the single homomorphism of unit rings $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; \backslash \; mathbb\; K$ is injective; as K is a body, it induces an injection of the body of the fractions of Z , namely the body of rational the Q (by definition of rational). In the second case, the single homomorphism of unit rings $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; \backslash \; mathbb\; K$ induces an injection of Z/pZ in K . However, as p is first, Z/pZ is a Corps finished; it is the single body finished F _p with p elements.

• Any body finished has as a cardinal a power of a prime number, which is its characteristic.

If K is a finished body, for reasons of cardinality, it cannot contain a copy of Q . By what precedes, it is of finished characteristic p and thus contains a copy of the body F _p . In fact, K is a vector space on F _p ; its dimension is necessarily finished. Thus its cardinal is p with the power his dimension.

• There exist infinite bodies having a nonnull characteristic $p$, this one being a prime number.

It is the case for example body of the rational functions on $\backslash \; mathbb\; Z/p\; \backslash \; mathbb\; Z$ or of the algebraic Clôture of $\backslash \; mathbb\; Z/p\; \backslash \; mathbb\; Z$.

• All Corps completely ordered has a null characteristic.

Indeed, single homomorphism $\backslash \; mathbb\; Z\; \backslash \; rightarrow\; \backslash \; mathbb\; K$ is increasing. Entire strictly positive is sent on a strictly positive element of the body, a fortiori different from 0.

It is thus the case of the bodies of the rational numbers $\backslash \; mathbb\; Q$, and thus of those of the real numbers $\backslash \; complex\; R$ and $\backslash \; mathbb\; C$ (since $\backslash \; mathbb\; Q$ is a subring of these two rings).

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