The ring associate same model

In a ring theory and abstract algebra, the ring associate same model (British: ring homomorphism) is a function to keep the structure between two rings.

Function f where the ring associate same model satisfies the following if R and S are rings when I write it properly: Is R → S [¹]; [²] [³] [⁴] [⁵] [⁶].

• For all former a and b of R, it is f(a + b) = f(a) + f(b)
• For all former a and b of R, it is f(ab) = f(a) f(b)
• f(1R) = 1_S.

(the identity element of an inverse element and the addition of the addition is a part of the structure, too, but it is not necessary to require them explicitly, and, as for the because condition, some of following properties do not consist of the condition mentioned above when this is because they follow it and drops one, condition f(1R) = 1_S.)

If R and S are rng (English version) (it is said with the ring of 擬環 and the non-unit), the natural concept [⁷] is the rng associate same model, and this is defined as a thing except the third condition f(1R) = 1_S by the above. I can think about the rng associate same model which is not the ring associate same model between the ring (of the unit).

The composition of two rings associate same model is the ring associate same model. I shoot the class consisting of all rings and in this way do an area as the を ring associate same model (area (English version) of the cf. ring). Particularly, I obtain a concept of the ring by oneself associate same model, the ring same model, the ring self same model.

Table of contents

Property

f: Direct next leaves from the definition when I assume R → S the ring associate same model.

• f(0R) = 0_S.
• It is f(-a) =-f(a) for all former a of R.
• For any unit a of R, f(a) is f(a^-1) = f(a) ^{-It is the unit that is 1}. Particularly, f guides the groups associate same model from group which the unit makes (multiplication) of R to the group which the unit of S (or im(f)) does (multiplication).
• Image im(f) of f is a subring of S.
• As for the nucleus of f, it is defined ker(f) = {a ∈ R as f(a) = 0}, and this is an ideal of R. A certain ring associate same model causes all ideals of commutation ring R in this way.
• It is the equivalent to be associate same model f being an injection and ker(f) = {0}.
• If f is bijection, inverse mapping f^-1 is also the ring associate same model. In this case f is called representation of the same type, and it is said that ring R and S are the same model. From the viewpoint of the ring theory, I cannot distinguish the ring that is the same model.
• Ring associate same model f: If there is R → S, 標数 of S finishes falling below 標数 of R. This may be usable to show that ring associate same model R → S cannot exist between a certain ring R and S.
• If R_p is the smallest subring included in R, and S_p is the smallest subring included in S; all ring associate same model f: R → S is ring associate same model fp: I derive R_p → S_p.
• If R is a body, and S is not a zero ring, f is an injection.
• If both R and S are bodies, im(f) is a subfield of S. So S of the body expansion of R can see it.
• If R and S are commutation rings, and S is an integral domain, ker(f) is a bare ideal of R.
• R and S are commutation rings, and S is a body; f all; shoot it, and であれば, ker(f) are very large ideals of R.
• f all; if shoot it, and で, P are ker(f) ⊆ P in a bare (very large) ideal of R, f(P) is a bare (very large) ideal of S.

Furthermore,

• The composition of the ring associate same model is the ring associate same model.
• 恒等写像 is the ring associate same model (the zero representation is not so).
• That is why all rings and class consisting of the ring associate same model form an area, the area (English version) of the ring.
• For all ring R, only ring associate same model Z → R exists. As for this saying, the integer ring is that it is targeted for the beginning in the area of the ring.
• For all ring R, only ring associate same model R → 0 exists; but 0 expresses a zero ring (the ring that the only cause is 0). As for this saying, the zero ring is that it is 終対象 in the area of the ring.

Example

• Function f which was defined by f(a) = [a_]n = a mod n: Z → Z_n all; shoot it, and the nucleus is nZ with the ring associate same model (see a congruence equation).
• Function f which is defined by f([a_]6) = [4a]₆: The nucleus that Z₆ → Z₆ is the rng associate same model (the winning rng self associate same model) is 3Z₆, and the image is 2Z₆ (as for this, of the same type with Z₃).
• Ring associate same model Z_n → Z does not exist for n≥1.
• Representation C → C taking the complex conjugate is the ring associate same model (in fact, it is a ring self-example of the same type).
• If R and S are rings, it is the equivalent that zero representation from R to S is the ring associate same model and that S is a zero ring. On the other hand, it is the rng associate same model for zero representation Hatsune (otherwise 1_R does not go to 1_S.).
• Function f which is defined by f(p) = p(i) if R[X] has a coefficient in real number field R, and a variable expresses the ring consisting of all the multinomial expressions of X, and C expresses a complex number body: R[X] → C (substitute imaginary unit i for variable X of multinomial expression p) all; shoot it, and is the ring associate same model. The nucleus of f consists of all multinomial expressions of R[X] which is divisible in X2 +1.
• f: If R → S is the ring associate same model between commutation ring R and S, f guides ring associate same model M_n(R) → M_n(S) between the line ring.

Area of the ring

The details refer to ":en:Category of rings"

The self-associate same model, same model, self- of the same type

• The ring by oneself associate same model (ring endomorphism) is the rings associate same model from a ring to oneself.
• The ring same model (ring isomorphism) is the ring associate same model with the both sides inverse element that is the ring associate same model. It can prove that it is the equivalent that the ring associate same model is the same model and that a stand is bijection for a function on gathering. If the ring same model exists between two ring R and S, R and S are called (isomorphic) of the same type. The ring that is the same model does not lay the difference that only changed the original name. An example: Except a difference of the same type, four rings of order of magnitude 4 exist. On the other hand, there is 11 in rng of order of magnitude 4 except the same model (as for this meaning it, a ring of which four orders of magnitude 4 that two are not the same model exists, and the ring of all other orders of magnitude 4 is that it is the same model in one of them.).
• The ring self same model (ring automorphism) is the ring same model from a ring to oneself.

With the injection associate same model all; shoot it, and the associate same model

I shoot it, and, as for the injection ring associate same model, the thing in the area of the ring is the same as (monomorphism). It is f: R → S shoots the thing which is not an injection, and であれば, a certain r₁ and r₂ are sent to the same material of S. Let's regard g₂ as two representation g₁ from Z[x] which represents x in r₁ and r₂ each to R. f∘g₂ is the same as f∘g₁, but f shoots the thing, and なので, this are impossible.

However, all; shoot it, and shoot it, and, as for the ring associate same model, Eppie in the area of the ring is totally different from (epimorphism). For example, inclusion Z ⊆ Q shoots Eppie of the ring; であるが all; shoot; ではない. However, all; shoot it, and the ring associate same model is just identical to strong epimorphism.

Footnote

1. ^ Artin, p. 353
2. ^ Atiyah and Macdonald, p. 2
3. ^ Bourbaki, p. 102
4. ^ Eisenbud, p. 12
5. ^ Jacobson, p. 103
6. ^ Lang, p. 88
7. ^ Hazewinkel et al. (2004), p. 3. Warning: They use the word ring to mean rng.

References

• Michael Artin, Algebra, Prentice-Hall, 1991.
• M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.
• N. Bourbaki, Algebra I, Chapters 1-3, 1998.
• David Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1995.
• Michiel Hazewinkel, Nadiya Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
• Nathan Jacobson, Basic algebra I, 2nd edition, 1985.
• Serge Lang, Algebra 3rd ed., Springer, 2002.

Allied item

• Coefficient expansion (English version)

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