CORRIENTES PARADIGMÁTICAS DE LA EDUCACIÓN ARTÍSTICA

In algebra, given aring homomorphismf : R→ S , there are three ways to change the coefficient ring of amodule;

namely, for a right R-module M and a right S-module N,

• f!M = M⊗RS, theinduced module.

• f_∗M =HomR(S, M ), thecoinduced module.

• f^∗N = N_R, therestriction of scalars.

They are related asadjoint functors:

f! :ModR⇆ ModS : f^∗ and

f^∗:ModS ⇆ ModR: f_∗. This is related toShapiro’s lemma.

43.1 See also

• Six operations

43.2 References

• J.P. May,Notes on Tor and Ext

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Characteristic (algebra)

Inmathematics, the characteristic of aringR, often denoted char(R), is defined to be the smallest number of times one must use the ring’s multiplicativeidentity element(1) in a sum to get theadditive identityelement (0); the ring is said to have characteristic zero if this sum never reaches the additive identity.

That is, char(R) is the smallest positive number n such that

1 +· · · + 1

| {z }

nsummands

= 0

if such a number n exists, and 0 otherwise.

The characteristic may also be taken to be theexponentof the ring’s additive group, that is, the smallest positive n such that

a +· · · + a

| {z }

nsummands

= 0

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (seering), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to thedistributive lawin rings.

44.1 Other equivalent characterizations

• The characteristic is thenatural numbern such that nZ is thekernelof aring homomorphismfrom Z to R;

• The characteristic is thenatural numbern such that R contains asubring isomorphicto thefactor ringZ/nZ, which would be theimageof that homomorphism.

• When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n · 1 = 0. If nothing

“smaller” (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial ordering because of such facts as that char A × B is theleast common multipleof char A and char B, and that no ring homomorphism ƒ : A → B exists unless char B divides char A.

• The characteristic of a ring R is n ∈ {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a ∈ R implies n is a divisor of k.

The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language ofcategory theory, Z is aninitial objectof thecategory of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).

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44.2 Case of rings

If R and S are rings and there exists aring homomorphismR → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is thetrivial ringwhich has only a single element 0 = 1. If a non-trivial ring R does not have anyzero divisors, then its characteristic is either 0 orprime. In particular, this applies to allfields, to allintegral domains, and to alldivision rings. Any ring of characteristic 0 is infinite.

The ring Z/nZ of integersmodulon has characteristic n. If R is asubringof S, then R and S have the same char-acteristic. For instance, if q(X) is a primepolynomialwith coefficients in the field Z/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since thecomplex numbers contain the rationals, their characteristic is 0.

It is called theFrobenius homomorphism. If R is anintegral domainit isinjective.

44.3 Case of fields

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or a field of positive characteristic.

For any field F, there is a minimalsubfield, namely the prime field, the smallest subfield containing 1F. It is isomorphic either to therational numberfield Q, or a finite field of prime order, Fp; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of thecomplex numbers(unless they have very largecardinality, that is; in fact, any field of characteristic zero and cardinality at mostcontinuumis isomorphic to a subfield of complex numbers).^[1]The p-adic fieldsor any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic p^k, as k → ∞.

For anyordered field, as the field ofrational numbersQ or the field ofreal numbersR, the characteristic is 0. Thus, number fieldsand the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. The finite fieldGF(pⁿ) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of allrational functionsover Z/pZ, thealgebraic closureof Z/pZ or the field offormal Laurent seriesZ/pZ((T)). The characteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.^[2]

The size of anyfinite ringof prime characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be avector spaceover that field and fromlinear algebrawe know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pⁿ. So its size is (pⁿ)^m= p^nm.)

44.4 References

[1] Enderton, Herbert B.(2001),A Mathematical Introduction to Logic(2nd ed.), Academic Press, p. 158,ISBN 9780080496467.

Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately.

[2] “Field Characteristic Exponent”. Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.

• Neal H. McCoy (1964, 1973) The Theory of Rings,Chelsea Publishing, page 4.