Projection coating

In mathematics, I say all things that I shoot it, and a nucleus becomes `smallest' among groups of associate same model representation P → M to projection module P and module M with the projection coating (しゃえいひふく British: projective cover).

Table of contents

Motive

Of projection module P where there is any module M all; shoot it, and is an associate same model image [¹].

${\displaystyle p\colon P\twoheadrightarrow M}$ 

Therefore, than an associate same model theorem

${\displaystyle P/\ker(p)=M}$ 

である. Therefore I choose module M so that ker(p) becomes `smallest' and say `approximation' thing which did it with projection coating in projection module P. More exactly for any partial module N of M

${\displaystyle N+\ker(p)=M\implies N=M}$ 

But, when is managed; p: It is said that P → M is projection coating.

Definition

I assume R a ring having an identity element, and, as for all the modules, shooting it left R module decides to point all to the associate same model of the left R module in the following.

That partial module K of module M is a surplus partial module of M (superfluous submodule); for part module N of any M

${\displaystyle N+K=M\implies N=M}$ 

But, I say that it is managed. Again all; shoot; ƒ: When nuclear ker(ƒ) of M → N is a surplus partial module; ƒ a surplus all; shoot it, and it is said that is (superfluous epimorphism). To projection module P and module M all; shoot

${\displaystyle p\colon P\twoheadrightarrow M}$ 

That Class の (P, p) is projection coating; p a surplus all; shoot; であることをいう. It is said that P is projection coating in this; and p: It may be said that P → M is projection coating.

Property

Uniqueness

Generally, the projection coating of the module may not exist [²]. (^[3] existing about the module in the アルティン ring.) However, I understand that it is decided uniquely if I exist from the next lemma.

Lemma [¹]

p: It is said that P → M is projection coating. Q is a projection module; all; shoot; q: If there is Q → M, part module R of ker(q) becoming Q = P⊕R exists; and limit q| _P: P → M is projection coating.

Naokazu

p_i: If P_i → Mi (1≤i≤n) is projection coating; (⊕p_i): ⊕P_i → ⊕M_i is projection coating, too [⁴].

Projection coating of the existing about module

I do P with the projection module which is not zero. With projection module P existing; about all quotient modules which are not zero of P as for the necessary and sufficient condition that is projection coating of the module direct; existing; is a thing about [⁴].

Allied item

• Introduction envelope
• The projection resolution
• Complete ring

Footnote

1. ^ ^{a b} Anderson & Fuller 1992, p. 199.
2. ^ Anderson & Fuller, p. 203.
3. ^ Iwanaga & Sato 2002, p. 253.
4. ^ ^{a b} Anderson & Fuller 1992, p. 203.

References

• Anderson, Frank W.; Fuller, Kent R. (1992). Rings and Categories of Modules. Graduate texts in mathematics. 13 (Second ed.). Springer-Verlag. ISBN 0-387-97845-3. 
• For Iwanaga, 恭雄, Sato, 眞久, 佐藤眞久 "homology algebraic theory Japan criticism company of a ring and the module", 2,002 years, it is the first edition. ISBN 4-535-78367-5。

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