文档库 最新最全的文档下载
当前位置:文档库 › $IF$环和拟$ZIF$环

$IF$环和拟$ZIF$环

392Vol.39,No.2 19963ACTA MATHEMATICA SINICA Mar.,1996

IF ZIF

(210098)

:ZIF IF ZIF

IF ZIF ZIF IF

IF ZIF?-

1

R.R.Colby[4]IF()IF()R

IF IF IF IF QF

[1,4?7].IF

IF IF IF

IF ZIF ZIF IF ZIF IF

2IF

2.1R M?0-I

f∈Hom R(I,M),g∈Hom R(R,M),g|I=f.

2.2([4],Theorem2[1]Theorem2.4)R

(1)R IF

(2)R R R R R?0-

(3)?0-R R

2.3R IF A(?nitely presented)R Ext n R(A,F) =0R F n>0

R IF 2.2,R 2.2[9]Lemma3.1

2.4R M?0-Ext1R(R/I,M)=0

I

19944281994929

2IF ZIF227 3ZIF

IF

3.1R()ZIF()R M,M()?0-

M?()R ZIF ZIF(

M?=Hom(M,Q/Z))

3.2R ZIF(Mα)α∈A R

(1)Mα?0-

(2)

α∈A

Mα?0-

(3)

α∈A

Mα?0-

(4)

α∈A

M?α

(5)

α∈A

M?α

(1)?(2);(1)?(3).[1]proposition1.8.

(3)?(4).[2]Theorem2.4,

α∈A Mα

?

~=

α∈A

M?α.R ZIF

α∈A Mα?0-

α∈A

M?α

(1)?(5).R ZIF Mα?0-M?α

[2]Theorem3.45,M?α

α∈A

M?α

3.3R E RτI:E??R I→(Hom R(I,E))?,f?a→

τI(f?a),f∈E?,g∈Hom R(I,E),τI(f?a):g→f(g(a)).(·)?=Hom(?,Q/Z),I

R

(1)E?0-R I,τI

E?

(2)E?E?0-

0→I i→R,I R

E??R IτI?→(Hom R(I,E))?

1?i↓↓(i?)?

0?→E??R RτR?→(Hom R(R,E))?

(1)E?0-(i?)?[2] 3.60τR

τI1?i1?i E?

(2)E?1?i[4]2(a),(c)

τI(i?)?I,Hom R(R,E)i?→Hom R(I,E)→0E?0-

3.4R R ZIF?0-R E

I,

σI:E??R I→(Hom R(I,E))?

228

39

3.3[4]

2(a).

3.5

R

()

R

()

ZIF

R

()

R

I

[2]

3.60

()R

E

σI :E ??R I →(Hom R (I,E ))?

3.4

R

()

ZIF

3.6

R

(1)R ZIF

(2)

R -M ,M ?0-

M ??

(1)?(2).R

ZIF

M ?0-M ?

[2]

3.52M ?

M ??

(2)?(1).M

R

M

?0-M ??

M ??

M ?

3.7

R

R

N

?0-

R

M ,

M ?=N .

R

ZIF

R

{N j |j ∈J }

R

?0-R {M j |j ∈J }

j ∈J ,M ?

j =N j ;

j ∈J

N j =

j ∈J

M ?

j =

j ∈J

M j ?

.

R

ZIF

3.2

j

N j

R

3.8

R

R

N

?0-

M ,

M ?=N ,

R

R

ZIF 3.9

R

ZIF

P

F

P R

F T =F/P

T

≤1.D

B

Ext 1R (T,D )=0.

B

R -D

B

B →D →0

0→D ?→B ?

(

(·)?=Hom (?,Q /Z )).

R ZIF

B ?

P

T ≤1.

Tor R 1(D ?

,T )=0.

T

0→K →F →T →0,

K

F

0=Tor 1R (D ?,F )→Tor 1R (D ?

,T )→

D ??R K →D ??R F →D ??R T →0t 1↓t 2↓t 3↓t 4↓

0=Ext 1(F,D )?→Ext 1(T,D )?→Hom R (K,D )?→Hom R (F,D )?

→Hom R (T,D )?→0

t 3,t 4

(

[8],P120),

[4]

2(a),t 2

t 1

Ext 1(T,D )?=Tor 1R (D ?

,T )=0

2IF ZIF229 Ext1R(T,D)=0.

4IF ZIF

4.1R IF R ZIF

2.2

3.5

4.2R IF0→A→E→B→0R

(1)?0-?0-

(2)

4.1,R ZIF

(1)A,E?0-A,B?0-0→A→E→B→0

0→A??→E??→B??→0. 3.6,

E??~=A??⊕B??.

3.6

E,B?0- 2.2,E,B

0→B?→E?→A?→0,

[2] 3.5.2,B?,E?A?A?

(R IF).A?0-

(2)(1) 2.2(3).

4.3R

(1)R IF

(2)R M,M?0-M??0-

(3)R ZIF R?0-

(4)R ZIF R M,M M?

(5)R ZIF R R,R R?0-

(1)?(2).M R M?0-R IF

2.2,M M??0-M??0-R IF

M? 3.3M?0-

(2)?(3).M?0-M??0-M???0-

M?

M?M???0-(2)M??0-

(2)M?0-R ZIF

M R M?(2),M?0-

(3)?(4).M R(3)M?0-R ZIF

M?

M?(3)M??0-M

(4)?(1).M?0-R R ZIF M?

(4),M 2.2R IF

23039

(3)?(5).

(5)?(3).M R

0→K→

Rλ→M→0,

K=Kerλ.

0→M?→

R?→K?→0.

M?

R?~=M?⊕K?. 3.2

R?(R R,R R?0-).M? 3.3(2)M ?0-

[1]Daminao R F.Co?at rings and modules.Paci?c J Math,1979,81:349–369.

[2]Rotman J J.An Introduction to Homological Algebra.New York-San Francisco-London:Academic Press,

1979.

[3]Anderson F W,Fuller K R.Rings and Categories of Modules.Graduate Text in Math,Springer-Verlag,

1974.

[4]Colby R R.Flat injective modules.J Algebra,1975,35:239–252.

[5]Jain S.Flat and F P-injectivity.Proc Amer Math Soc,1973,41:437–442.

[6]Ahsan J,Lbrahim A S.Some homological characterizations of reguler https://www.wendangku.net/doc/321980929.html,munications in Algebra,

1982,10:887–912.

[7]Matlis https://www.wendangku.net/doc/321980929.html,mutative semi-coherent and semi-regular rings.J Algebra,1985,95:343–372.

[8]Cartan H,Eilenberg S.Homological Algebra.Princeton,1956.

[9]Stenstr¨o m B.Coherent rings and F P injective modules.J London Math Soc,1970,2.

IF Rings and Quasi-ZIF Rings

Zhu Xiaosheng

(Department of Mathematics and Physics,Hohai University,Nanjing210098,China)

Abstract:In this paper,Quasi-ZIF rings are de?ned.The structure of IF rings and Quasi-ZIF rings is studied.Furthermore,the relations between IF rings and Quasi-ZIF rings are discussed.Finally,IF rings are represented with Quasi-ZIF rings.

Keywords:IF ring,Quasi-ZIF ring,?0-injective,?at,character module

相关文档