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-
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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.
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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