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1 /46Φ/42 c ψ E Vol 46 No > ADVANCES IN MATHEMATICS CHINA July 217 doi: /sjz215154b χ 'ffiμ Φnrff fi"# UH 1 [BK 2 Q]G 1 1 ff^s&k_χcfi ff^ _ 239; 2 3#χbχΞ vf 2196 $ψ: 6F<MV>PJDE/A:5^W9@ -RSOV>CTL? ; F IPJD56 F< 6F<81CTL? ;IPJD5E/A: 43C F D2ZPJ75WN u}l: PJD; 6F<; PJ7 MR21 +»s : 13C1; 13C6 / *νs v: O1533 ßffj Π: A ß%iv: ]9%g}b 1976 [2]!Λ]flNAj6S*' Z$+] i WDwΠL] J ] 2 NA] flw%fl@6»ο+» us@ Artin fifl5_mffiilgj>"+!λ &4V[K6'' q pcω!λ ~+u3λv T 1999 Haghany W Varadarajan [4 5] 6 WDwΠL]~νοΛv+!Λ e Krylov W [7] Λv!Λο WΠL]g3%+ΩP p P8 WΠL]+ΩPe»!Λ Sfi r [3 1 12] 6b}M Kρο WΠL]GwJ%Wfl%+* fl[ ±"[ $2+;±QU P8 WΠL]+ffΠ Ifi r Φ [1 6 9] ΩP +lv r [7] +]Ψ}U5% <+z] %Ψ}Uh<+f% 1 ffi &wfl m~t C6 WDwΠL]g~'ffl+ F V3 5A± W: { A N a n F a A b Bm Mn N} M B m b 3 A B [5% +z] M [f B-6A- d% N [f A- 6 B- d% om ϕ : M A N B ψ : N B M A [d%uk 76?+ m m M nn N ffid ϕm nm mψn m ψn mn nϕm n 2 F +n<wμ<a±: a n a n + a + a n + n m b m b m + m b + b a n a n aa m b m b + ψn m an + nb ma + bm bb + ϕm n B F 68i2+?hg ] JJ]Ξ[ WΠL] %»} Morita Λv] aqπl] I@2: c fl: _Q:s[1fl No KJ216A545 _QG c>ffχc No KJ215B12 uqb@163com

2 558 b χ D 46Φ ϕ ψ +ffiaff[] A B + fi J fi%»}] F +e fi trace ideal mfl+ W ΠL]Ψ 5ρe fi 42*Q WDwΠL][ffi ]+'n» i]q C'Jß +] K ΞU@ρ!-< e e 1BB]uN8 WΠL] eke ek1 e 1 eke 1 ek1 e C U V [] K G+5ΨO f 6K % Bbuk] EndU V un8 WΠL] EndK U Hom K VU Hom K U V End K V 4ffifirP8 WΠL]+ΩP p@]fl9%fl!λ 5Z$+ L8+P8 WΠL ]+»afi r [3] ±" I WΠL] F M A N B G%+zN ψfitt A- W B- % NA F - % M [f A- % [f B- % f : N B [ A- %uk g : M A [ B- %uk 6?+ m M n N y 5 nm nm mny mny 3 m ny +%μ2}: a n m b y a + ny m + by Bßfflν< fig f F - % C 7 N M B [ +a% P8 F%+NAfi r Φ [7] R χjf F - %%fiω Gßfflν+ W ff V [f F - % e 1 1 e 1 B ev [f A- % 1 ev [f B- % Bf F - % V fiω} ev 1 ev HG`>< ttfνdfiq%4f A- % NAρf F - % 3%μ2}: B M A f : N B M A N B M A A A g : M A M A [f F - % e 68f B- % N B %[f F - % 2 ffi &wfl ffl mqωz 6b/ [!ΛwJ%* fl[+z$]5 6 R- % P ψ» Hom R P R [ P +6 μjuψf y i P ϕ i Hom R P R»} P +6b C3ffid±μr: 1 6 P?+'J<g ϕ i e855ψj ϕ i V ϕ i p 2 p ϕ ip ϕ ipy i ±"Kρ6b/ 3P#fi r [ ] 'J%[wJ%'7}'j5'J6b oλ 21 pψy!λ wj%+6b9b4%+b5 W+ffiu wj%+<g26bωρs 3Ω< ffi ' 9b4%+χJ<g2bΩρS 3Ω< ' ±"tt% +6b NA F - % M +6b A M {ϕ i i } [wjf A- % +6b { i } [ F - % M +O <ψ {ϕ A i 1 A N ϕ i } [ Hom F M A M B +'J<gψ 3 ϕ i : A 1 ϕ i : M A M A A M

3 42 fi9fi GffiP O: μ±m^hk&7/ch ?+ i I 5 1 ϕ i m mϕ i fi r [12] ±"P#O <ψ { i } WkJψ {ϕ i 1 ϕ i } N F - % oλ 22 [ F - % M +6b A 2 M [wjf B- % C {ψ j y j } j J [ +6b B { 1 ψ j ψ j yj }j J [ F - % N B +6b M +6b A 1 M [wjf A- % C {ϕ i i } [ +6b B {ϕ i 1 ϕ i i } ± VffP# 1 efip# 2 #P6?+ ϕ i 1 ϕ i m i m A M 5 A m A ifi ZTG i ϕ i 1 ϕ i m ϕi i mϕ i ϕ i i ϕi i mϕ i i mϕ i i mϕ i i m ϕ i i m 2 *P 2 22 fi2 NA WΠL] F M A N B GwJ% Q P [8] Krylov 9 ardykov P#ο P N B Q M A 42 22Q {ϕ i 1 ϕ i i } W { 1 ψ j ψ j y j } Aff[ F - % N B j J M A 9 +6b 9fi2 P Q +O <ψw6bkja±: { pt q t } t T { pi } q i { } pj q j j J { i { { ϕi 1 ϕ α t β t i t i I 1 ψ j ψ j t j J 3 T I J οξ 21 {α t β t pt q t } t T [ F - % P Q +6b ± 6?+ p q P Q Vff#P p α t β t q t T?H [4 2 32] +P#fiAμ8ffl: pt q t } t i I yj} t j J; p q

4 56 b χ D 46Φ ^ ' p q 1 ' t I S 2 ' t J S α t β t m S m ϕ i 1 ϕ i α t β t m 1 ψ j ψ j α i β i m m α j β j m m ϕi mϕ i ; 9 p pt α t β t pt q q t β t t m t T t Tα ϕi q t mϕ i ϕi i ϕi i mϕ i mϕ i i ϕ i i mϕ i i m efip#' p q α t β t n y y S 5 n y pt y q t n y y pi q i 2 22 Wyfl 21 +'JTy2A±: oλ 23 1 [wj A- %'7}' M A [wj F - % 2 [wj B- %'7}' N B [wj F - % ± 46b/ Wyfl 21 * ""q ο [8 2 31] +P#Tν 681l%%5 e+yfl P# r [7 $p 63] Krylov 9 [8] Kρο F - % P Q [wj%wfl%+οaφ$rs ΩP+ zfl@z"+p#`2 oλ 24 [8] 6 F - % P Q ±Dr-p: 1 Q P [ F - wj% 2 P/NQ [ A- wj% Q/MP [ B- wj% 7 N Q/MP NQ M P/NQ MP 3!@ A- wj% B- wj% V* P NQ Q MP 7 M A MP N B NQ

5 42 fi9fi GffiP O: μ±m^hk&7/ch A- wj% B- wj% V* P Q M A N B i]q?+%45*jξ &flffi[i5+%45wje ± mfl WΠL]G% +wje+!@ p 'J]»}f 6 z;] ASB]G+χJ1l%4[wJ% z;] G+χJ%45wJE ±"Kρ F - % 5wJE+'Jο$rs 5wJE+οAΦ$rs[ A- % oλ 25 M F [ WΠL] [ F - % B /N B- % /M Aff5wJE ± οaο om A- % /N B- % /M Aff5wJE σ 1 : P 1 /N σ 2 : P 2 /M ff η 1 : /N η 2 : /M [buk 4 P 1 P 2 [wj%q!@%uk γ 1 : P 1 γ 2 : P 2 V* η 1 γ 1 σ 1 η 2 γ 2 σ 2 ff ε 1 :N B P 2 P 1 3 ε 1 N B P 2 f1 N γ 2 ε 1 P1 γ 1 ff ε 2 :M A P 1 P 2 3 ε 2 M B P 1 g1 M γ 1 ε 2 P2 γ Q [ F - wj% ±"P# N B P 2 P 1 M A P 1 P 2 ε ε 1 ε 2 : +wje }ffivff#p±μ: P1 M A P 1 N B P 2 P 2 N B P 2 P 1 M A P 1 P 2 [ F - % 1 ε 1 : N B P 2 P 1 ε 2 : M A P 1 P 2 [ffiuk ZTG } η 1 ε 1 N B P 2 P 1 η 1 N + ε 1 P 1 + σ 1 P 1 /N i N + ε 1 P 1 χ9 ε 1 [ffiuk u fip ε 2 [ffiuk 2 l K kerε 1 L kerε 2 B K N B P 2 P 1 L M A P 1 P 2 4 [7 $p 32] V$P# N B P 2 +K P 1 P 1 M A P 1 +L P 2 P 2 4 ε 1 N B P 2 N * ε 1 1 NN B P 2 + K 7} ker σ 1 γ1 1 NN B P 2 + K P 1 i N B P 2 +K P 1 P 1 u fip M A P 1 +L P 2 P 2 Φ$ο fiom ε ε 1 ε 2 : N B P 2 P 1 M A P 1 P 2 [ F - % +wje 3 P 1 P 2 [wj% ff K kerε 1 L kerε 2 B ε 1 :N B P 2 P 1 ε 2 :M A P 1 P 2 [ffiuk 7 K N B P 2 P 1 L M A P 1 P 2 ε 1 N B P 2 4 [7 $p 32] N B P 2 +K P 1 P 1 M A P 1 +L P 2 P 2 7} f1 N ε 2 P2 i ε 1 N B P 2 fn B N ε 1 1 fn B N B P 2 +K

6 562 b χ D 46Φ ff i 1 : P 1 N B P 2 P 1 }5Buk B σ 1 η 1 ε 1 i 1 : P 1 /N [ A- %ffiuk fl7 ker σ 1 ε 1 1 fn B P 1 N B P 2 +K P 1 χ9 ker σ 1 P 1 O σ 1 : P 1 /N [ A- % /N +wje efip B- % /M 5wJE σ 2 : P 2 /M οξ 22 M F [ WΠL] B±"μr-p: 1 F [fz;]; 2 A B Aff[fz;] ± M F [fz;] :1l A- % B M A [1l% 7 F [fz;]q M A [wj% 4 [7 2 73] Q [wj% χ9 A [fz;] u fip# B %[fz;] R A B Aff[fz;] 4 [7 2 73] Q?+1lf F - %[wj% 9 F [f z;] fl gif NH_mio+ΠRßCt-! k ρfi [1] Auslander M Reiten I and Smalø SO Representation Theory of Artin Algebras Cambridge: Cambridge Univ Press 1997 [2] Goodearl KR Ring Theory: Nonsingular Rings and Modules New ork: Marcel Dekker 1976 [3] Green EL and Psaroudakis C On Artin algebras arising from Morita contets Algebr Represent Theor : [4] Haghany A and Varadarajan K Study of formal triangular matri rings Comm Algebra : [5] Haghany A and Varadarajan K Study of modules over formal triangular matri rings J Pure Appl Algebra : [6] Kasch F and Wallace DAR Modules and Rings London: Academic Press 1982 [7] Krylov PA and Tuganbaev AA Modules over formal matri rings J Math Sci : [8] Krylov PA and ardykov E Projective and hereditary modules over rings of generalized matrices J Math Sci : [9] Lam T Lectures on Modules and Rings Grad Tets in Math Vol 189 New ork: Springer-Verlag 1999 [1] Tang GH Li CN and Zhou Q Study of Morita contets Comm Algebra : [11] Tang GH and Zhou Q A class of formal matri rings Linear Algebra Appl : [12] ardykov E The dual basis of projective modules over generalized matri ring Bull Tomsk State Univ 27 3: The Dual Basis of Some Projective Modules Over a Formal Matri Ring and Its Application U Qingbing 1 ZHANG Kongsheng 2 WANG Zhengping 1 1 Department of Basic Courses Chuzhou Institute of Technology Chuzhou Anhui 239 P R China; 2 Department of Mathematics Southeast University Nanjing Jiangsu 2196 P R China Abstract: The dual basis is an important tool for studying projective module internal structure In this paper the projective module over the formal matri rings are characterized by the dual basis Also the sufficient and necessary conditions are found for an F module to admit a projective cover Keywords: projective module; dual basis; projective cover

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