TAP CHI KHOA HOC SO 16 3 www.htu.edu.vn
MOT SO VI DU VA TEVH CHAT CUA MODUN KHOfP LS Van An
Trudng Dai hoc Hd TTnh Email: [email protected]
NguySn Thi Hai Anh Truemg Bai hoc Hd TTnh Email: ank.ngi^[email protected] Ngdy nhgn bdi (received): 20/3/2019
Ngdy nhgn bdn sua (revised): 5/4/2019 Ngay nhgn dang (accepted): 21/4/2019
TOM T A T
Vanh R dugc gpi la vanh khop phai neu voi mpi R - modun con phai hmi han sinh F cua R°, moi dong cau (p.F >R co the ma rpng tdi dong ciu y/:R" >R. Khai niem vanh khop dugc dua ra trong cac bai bao D. Wilding (2013) va Y. Shitov (2014). Trong bai bao nay, chung toi gioi thieu khai niem modun khop va nghien cmi mpt so tinh chat cua lop modim nay.
Tir khoa: vanh khdp, modun tua npi xa, modun khop.
Someexamples and properties of exact modules Abstract
The Ring R is a right exact if, for every finitely generated right submodule F of^
and every right R homomorphism q>\F >if, can be extended to u right R - homomorphism yriR" ^R satisfying g)[x) = y/{^x) for any element x of F. The concept of The exact ring is given in articles ofD. Wilding (2013) and Y. Shitov (2014). In this paper, we define an exact module and identify some properties of these exact modules.
Key words: exact ring, quasi - injective module, exact module.
I. Dat v^n ah
Trong bai bao nay khai niem vanh dugc nhae den nhu la vanh ket hgp co don vi va cac modun la R - modun phai unitar. Cho cac R - modim phai M va N, N dugc gpi la M - ndi xg {M - injective) neu voi mpi modun con X ciia M, moi dong cau ^: X >N co the mdr rpng tdi dong cau ^:M >N. N dugc gpi la noi xg {injective) neu no la M - npi xa vdi mpi modun M. Modun N dugc gpi la tua ndi xg {quasi - injective) neu no la N - npi xa.
Dtnh nghia va tinh chat ciia lop modun tua npi xa co the tim thay trong S. H.
Mohamed (1990) (Chapter 1).
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Cho vanh A va G la mpt nhom khong nhit thiSt hdu han, vanh /! = ^[G] duc"^ S'' viath nhom {group ring) oia G tren A. Khi do cac phin tii ciia A[G] duoc ky hieu hinb thuc dudi dang r='^a^.g, trong do gsG.a^sA voi a^*0 tai hiru han gia tri. Cac phep'°
«<^
dugc xay dimg nhir sau:
trong do c^=Y^ ^A=Z''jrV'*-
Vdi cac phep toan nhu tren, vanh nhom A[G] la mot vanh kdt hgp co don vi. Dinh ngjiia va cac tinh chat cua vanh nhom co the tim thay trong I. G. Connell (1963) va D.
Farkas(I973).
Cho vanh R. Vanh R dugc gpi la vanh khdp phdi neu vdi mpi R - modun con phai huu han sinh F ciia R", moi dong cau ^:F >R cd the md rgng tdi dong cau y/ -.R" >R. Khai niem vanh khdp dugc dua ra trong D. Wilding (2013) va cac tac gia da chiing minh vanh tua npi xa phai la vanh khdp phai. Sau do trong Y. Shitov (2014), tac gia da chi ra mpt vi du ve vanh khdp phai nhung khong tua ngi xa phai. Do do cd the khang dinh rang vanh khdp la md rpng cua khai niem vanh tua npi xa.
Trong bai bao nay, chiing toi gidi thieu khai niem modun khdp va nghien cuu mpt so tinh chat cua ldp modun nay.
II. Ket qua
Dinh nghia 1. Cho cac R - modun phai M va N. N dupc gpi la Af - khdp {M - exact) neu vdi mpi modun con huu ban sinh F cua M", moi dong can ^:F >jv cd the md rpng tdi dong cau y/: M" >N. N dugc gpi la khdp {exact) neu no la N - khdp.
Menh de 2. Cho cdc R - modun phdi M vd N.
(i). Neu N IdM- ndi xg thi NldM - khdp.
(ii). Neu N Id modun tira noi xg thi N Id modun khdp.
Chung minh.
(i). Gia su N la M - npi xa, trudc het chung ta chung minh N la M" - noi xa TTi S H. Mohamed (1990) (Proposition 1.5), N la M, - npi xa vdi mpi tap ^^ so I khi va chi khi N la @M,.- npi xa. Chpn / = {l,...,n} vaM, =M vdi mpi ie{l,...,«}, tir gia thigt N la M-
16/
npi xa, chiing ta co N la M" - noi xa. Bay gid, xet F la modun con hiru han sinh cua MD ^ ddng ciu <p: F >N, tu dinh nghia ciia modun M" - npi xa, suy ra ^j co th6 mo rfi d6ng ciu if^: M" >N. Do do N la M - khdp.
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(ii). "S^ N la modun tua npi x^ nen N la N - ngi xa. Theo (i), N la N - khdp, suy ra N la modim khdp. n
Tiip theo chiing toi se xay dung mgt so vi dy de khang dinh ldp modun khdp la md rpng thirc su cua Idp mddun npi xa. Cho cac mien nguyen A va B vdi A la midn con cua B. Phin tir b ciia B dugc gpi nguyen tren A neu ton tai da thiic f(x) ciia A[x] sao cho / (b) = 0. Khi do tap cac ph§n tut ciia B nguyen tren A la mot miSn con ciia B chua A.
Ching han vdi cac mien so nguyen Z va mien so phiic C; tap hgp cac s6 dai s6 tren Z la mpt mien con ciia C va chua Z. Ky hieu tap cac phan tii nguydn ciia B tren A la A®
va ta gpi mien nguyen nay la bao ddng nguyen cua B tren A. Neu A la mpt midn nguyen vdi trudng cac thuong K, khi do A*^ gpi la bao ddng nguyen cua A. Ndu A^ = A, thi midn nguydn A dugc gpi la vanh ddng nguyen. Mien nguyen R dugc gpi la mien Dedekin {Dedefdn domain) ndu R la vanh Noether, ddng nguyen va mpi iddan nguyen td la idean cue dai. Mien Dedekin R khdng phai la trudng dugc ggi la midn Dedekin thirc su. Vanh so nguyen Z vdi trudng cac thuong la trudng cac so huu ty Q la vi du vd midn Dedekin thuc sir.
Cho M la modun trdn vanh giao hoan R va x la mpt phan tu ciia M. Ky hidu Ann[x):={aeR:ax = 0} dugc gpi la linh hda tic ciia x trong R. Nh$n xet ring Ann(x) la mpt iddan cua R. Phan tii x dugc gpi la khdng xodn ndu Ann ( J:) = 0 va trudng hgp ngugc lai thi X dupc gpi la phin tu xodn. Tap hgp cac phin tu xoan cua M la mot modim con va dugc gpi la modun con xoan ciia M va ky hieu la Mj. Neu M.^=M dii M dugc gpi la modun xoan va neu M^ =0 thi M dupc gpi la modun khdng xodn.
Cho M la modun t-en vanh giao hoan R, khi do idean nguyen td P ciia R dugc gpi la idean nguyen td lien kit cua M, ndu tdn tai phin tii m ciia M sao cho ^/in(m) =P. Ky hieu
ASS[M) la tap tat ca cac iddan nguydn to lidn kdt ciia M. Trdn midn Dedekin R, modun M ¥^0 trdn R xoin khi va chi Idii 0 g ASS[M) va M la mddim Idiong xoin khi va chi khi Ass ( M ) = 0. Modun M dugc gpi IkP- nguyen sa ndu Ass [M) = P. Modun con N cua M vdi N la P - nguydn so dugc gpi la thdnh phdn P ~ nguyen sa ciia M.
Cho R la mgt mien Dedekin vdi tnrdng cac thuong K. Gia su P la iddan nguydn to cua R, ky hidu R{P'^\ la thanh phin P - nguydn so ciia modun xoin K/R. Trong trudng hgp cu thd, neu R la vanh cac so nguydn Z vdi trudng cac Ibuong Q, thi Z ( P " j dugc gpi la cic Prufer nhdm Zf/?"jvdip la mdt sd nguyen to nao dd.
Gpi R la mpt midn Dedekin thirc su vdi trudng cac thuang ciia nd la K. Dat
^ = [ © ^ J @ [ @ 4 J @ [ © - S j , trong dd K,=K vdi mpi f e A , 4 s ( i ? ( / r ) \ ^ vdi
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mgi s^Q. va B^^[RIQ.) v d i m p i r e T , d d a y (P)^^ va ( g , ) . la cac idean nguyen t6 khac khong ciia R thda man P,^Qj cho mgi cap {i,j)eI^J. Theo D. K. Tutuncu (2010) (Proof of Theorem 3.4), M co su bidu dien dudi dang M^I(M)^S{M), trong do I(M) la modun con npi xa tdi dai ciia M va S(M) la modun con nua don nhung khong npi xa. Chpn M khong la modun npi xa khi do S{M)=N^O la modun khong tua noi xa.
Dinh ly 3. Cho R Id mien Dedekin thicc sir vai trudng cdc thicang K. Bgt
trong do K,^K vdi moi / e A , 4 ^ ( ^ ( i f ) ) v&i moi s&Clvd B^ = (R/Qj).^
vaimoi r^T (a day (P,).^ vd [QJ) la cdcidean nguyen td khac khdng cua R thda man P,^Qj chomoicgp (i,j)&lxj); I(M) la modun con noi xg tdi dgi cua M, S(M) Id modun con nica dan nhimg khdng ndi xg. Neu M khong la modun ndi xa khi do S[M)=N Id modun khdp nhimg N khdng la mddun tua ndi xg.
Chimg minh.
Dau tien theo phan tich d trdn chiing ta cd S{M)=N^Q. Chiing ta se chiing minh N la mddun khdp b i n g each chiing minh N la N - khdp. Gia sir F la modun con huu han sinh ciia N", vi N la mddun nua don ndn N° la nira don va F la hang tii true tiep cua N°.
h
Hon nua, F cimg la mddun mia don (va huu han sinh), dat F = @f^R, trong dd ff.R la modun con don ciia F. Nhan xet rang B = [f,f2,... ,f] la co sd huu han cua F. Gpi
^ : F >N la mpt dong cau, khi do ^ xac dinh vdi he sinh p { S ) = {^(_^),...,^(^J,)},
h
tiic la vdi mpi phan tir X cua F, chiing ta CO ^ = ^ / i ' ' t , trong dd rjei?,(fe=l,2,...,A) thi
h
<p{^)=ll9{fi,h-
Dat N" = F @ G, khi do vdi mpi phan tii m ciia N" cd sir bieu dien duy nhit dudi dang m = x+g, trong do xGF,g&G. Tir do suy ra m='^f^rf.+g. Xay dung d6ng ciu
h
y/:N" >N, xac dinh bdi (i'('n) = y ^ 9 ' ( / i ) ' i - Chiing ta se chiing minh y/ la md rpng
k=\
ciia (p, tiic la y/i = (p, trong dd i:F >N" la phep nhiing.
h
That vay, xet phan tu x bat ky ciia F, chiing ta cd x = ^^ f^r^., suy ra
*=i
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\k=\ ) k=\
Do do v(''(jc)) = ^(jc) vdi mpi phan td x cua F, hay yf la md rdng ciia <p. V^y N la N - khdp, suy ra N la modun khdp.
Luu y rang, N khong la R - modun phai t^a npi xa da dugc chi ra d trdn vdi dieu kien M khdng la mddun npi xa. •_!
Tidp theo chiing ta sd chi ra thdm mdt vi du ve mddun khdp nhung khdng phai la modun tua npi xa.
Cho G la mot nhdm, nhdm G dugc gpi la huu ban dja phuang ndu mpi nhom con huu ban sinh ciia G la nhdm hiiu ban. Dd xac dinh nhdm huu ban dja phuong chiing ta cd did xay dimg nhu sau:
Cho I la tap chi sd vd han cd ban sd ddm dugc hoac continum. Gpi (G^) la mdt hp bat ky cac nhom hiru ban va dat G = @ G, la mpt tdng true tidp cac nhom con Gi. Khi do G la nhom huu ban dia phuang. Gpi K la mpt tnrdng nao do, suy ra K tua npi xa phai (va trai) va do do K la vanh khdp phai (va trai). Xet vanh nhdm K[G], khi do chiing ta co:
Dinh ly 4. Cho G la nhdm hiru hgn dia phuang nhimg khdng la nhdm him hgn vd K Id mot trudng ndo do. Khi do K[G] la K[G] - mddun phdi khdp nhtmg khdng la mddun tua ndi xg.
Chung minh. Theo Y. Shitov (2014) (Theorem 3.5), K[G] la vanh khdp phai. Mat khac, G la nhom huu han dja phuong nhimg khong hiru ban ndn theo D. Farkas (1973) (page 314), K[G] khong la vanh tira ngi xa phai. Do dd K[G] la K[G] - mddun phai khdp nhimg khdng la modun tua npi xa. D
Tir cac Menh dd 2, Dinh ly 3, Dinh ly 4 chiing ta khang dinh rang ldp mddun khdp la md rpng thuc sir cua ldp mddun tua npi xa. Tidp theo chiing toi dua ra mgt so tinh chat ciia ldp mddun khdp. Chiing ta cd, mddun N dugc gpi la noi xg chinh {P - injective) ndu vdi mpi phan tir a ciia R, mdi dong cau ^:aR ^A'^ co thd md rpng tdi dong cau y/:R >N.
Dinh Iy 5. Cho cdc R - mddun phdi M vd N. Khi do:
(i). Neu NldM - khdp thi NldA- khap v&i bat ky mddun con A cua M.
(ii). Neu NldM- khdp thi N Id M/A - khdp v&i bdt ky mddun con A cua M.
(iii). Neu NldM- kh&p vd A la hang tti true tiep ctia N thi A ciing Id M- kh&p.
(iv). Neu A Id hgng tic tuc tiep cua NvdNld modun kh&p thi A ciing Id mddun kh&p.
(v). Neu NldR- kh&p thi N Id mddun ndi xg chinh.
(vi). Mddun WN^ IdM- khdp khi vd chi khi N,ldM- khdp v&i moi i e /.
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TAP c m KHOA HOC s o 16
(vii).M6aun ^N, laM-kh&p khi va chi khi N, la M-khap vai mgii^I- Chimg minh.
(i). Gia su N la M - khcSp va A la modun con bit ky cua M. Chiing » *'.^ ^i minh Nia A-khop. Xet Fla modun con hiru han sinh ciia A°va <p:F > ^ ' ^ " ' ; . cau, khi do F cung la modun con hiiu han smh ciia M". Vi N la M - khop nen *" "^
rongtaid6ngciu p ' : ^ / " >N.Bity/:=^'\:A'' >Ar 14 han chi cua tp Chiing ta CO !C la mo rong ciia tp tren A°. Do do N la A - khop.
(ii). Gia su N la M - khop va A la modun con bit ky cua M. Chung ta se chung minh N la M/A khop. Xet F la modun con huu han sinh cua (M/A) va g:(M I A)" > ^ A » la ding ciu, suy ra g(F) la modun con hdu han sinh cua y ^ , . Goi <p:F >N la mgt dong ciu bit ky, chiing ta chiing minh p co the mo rong toi dang ciu if\(MIA)" *N. Vi g(F) la modun huu han smh nen g(F)=(I,,7^ '7,) = [x,+A\x^+A\...,x,+A''). Goi F' = {K„ x^,..., x^) la mMim con him han sinh ciia M", de thay A° la modun con cua F' (vi A" la lop 0 ciia /.n )• '^^
phep chieu p-.M" ^M" I A" xac dinh boi
p :{{jn^,jn2,...,mj)\ = {m^,m^,...,m^^+A",
khi do p cam sinh dong cau p'= p\p.'.F' >g(F). Dat tp'=(pg~^p*:F' >i^
la dong cau xac dinh boi ^1 ^^.'i- = ? » § " ' ^ ^ f ? ; = ^ P g ~ ' ( ^ , )';•, trong do r^,r^,....r^&R. Vi N la modun M khop nen tp' co the mo rong toi dong cau 9 '-M" >Af, tile la (p\ (y) = cp'{y) = <p[g~' (p'{y))) voi y la phin tii bit ky ciia F' va i,.F' >M" la phep nhiing. Xay dung tuong ling p " : Af"/, ^N xac dinh boi
/ A
«7"((m„mj,...,m.)+^')=?j'((m|,nt,,...,m„)). Chiing ta chiing minh (p~ la d6ng ciu. That vay,xetx=(m„m2,...,m,)+^"=(m'„m'j,...,m'.)+^" voi hai phin tu dai dien khac nhau
m=(m„mj,...,ra.) va m'=(m'„m'„...,m'.), suy ram-m's^" c F ' . DiSu nay din dSn ip'[m + A')=<p'{m) va ^~(tn'+A")=(p'{tn'). Mat khac voi moi phin til y oila F' chiing taco:
'p{y)='p'h(y) = <p'{y)=<p{g~'ip'{y)))=9(g''ip{y)))=ip(g-'(y+^.\\
Dat y = m - m ' e ^ ' ' c z F ' , suyra
^'{y)=9'('"->n') = 'p{g'("•-"•'+'^"))=9{g~'i°]) = 0<^9 ('>')=^•f^^.^
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Ha.y^^(m+A")=grim'+A"), tiic la ^ " khdng phu thudc each chpn phln tu dai didn. Dd dang chiing minh ^** la mot ddng cau.
Cu6i cung, dat y/=<p'g:[M I A"\ >N, chung tacd yf la dong cau md rpng ciia g>. Vay N la M/A - khdp.
(iii). Gia sir N la mddun khdp va A la hang hi trite tidp cua N, chiing ta se chiing minh A la M - khop.
Dat N=A®B, ggi F la mddun con huu han sinh ciia M"* va (p:F >A la mpt ddng cSu, chiing ta chiing minh q) md rdng duoc. Gpi i^:A >N = A®B la phep nhiing, khi dd i^ip'.F >N la mdt ddng cau. Vi N la M khop ndn tdn tgi ddng ciu cp :M" >iV la md rdng cua i^q>, tiic la i^q>=<pi^ trong do i:F yAf" la phq) nhiing. Xet tuong iing y/:M" >A, xac dinh bdi y/ = p<p , vdi p:N = A®B ^A la phep chieu. Chiing ta chiing minh p =y/i. That vay, xet phan tir m ciia F, chiing ta co
yfi{m) = p(p'i{m)==pi,<p(m) = pi,{a)=p(a + 0) = a=<p{m)
dday ^(m)=ae.<4, suyra y/i = <p. Do dd y/ la dong cau md rpng ciia <p. Vay A la M-khdp.
(iv). Gia sii N la mddim khdp va A la hang tir true t i ^ cua N, chiing ta chiing mjnh A ciing la mddun khdp. That vay, vi N la N - khdp nen theo (iii), A la N - khdp. Mat khac vi A la mddun con cua N ndn theo (i), A la A - khdp. Vay A la mddun khdp.
(v). Gia sir a la phan td cua R va ^-.oR >N la mpt dong cau. Khi do aR la mddun con cyclic (do dd huu ban sinh) ciia R", va bdi N la R - khdp, <p cdHoh md rpng tdi ddng cau ^ : R" ^N. Dat y/ := ^ ' |^ : R ^N la han chd ciia g>' tren R. Chiing ta cd y/ la md rdng ciia ^ trdn R. Do do N la mddun ndi xa chinh.
(vi). Ndu Y[^i 1^ M - khdp, thi theo (iii), Nj la M - khdp vdi mpi I G / . Gia sir ngugc lai, N, la M - khdp vdi mpi / e / , chiing ta chiing minh YL^i 1^ M - khdp. Gia su F la modun con huu ban sinh cua M" va <p:F ^^IT-'^, ^^ ^^Pt dfing cau. Xet Pj'Y\.^i ^^j la pttep chidu tu nhien vdi mgi y e / . Khidd Pj^.F >Nj la ddng can. Tir gia thidt Nj la M - khdp, suy ra ton tai cac ddng can y/ iM" •A', la md rpng ciia Pj^ vdimpi jel. Dat y/:M" ^ n ^ * xac dinh bdi
IS/
Chiing ta cd (c la mpt ddng can. Chung ta se chiing minh y/ la md rgng ciia tp trdn M". That vay, gpi fi'.F >M" la phep nhiing, suy ra p (p=y/ fi vdimpi y e / . Didu
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TAP c m KHOA HOC SO 16 10 i w * ^ " ^
u'- ^°
nay din d8n ¥M{'rt) = iy{m) = Y,fi('")=Y.'l'A"') = llP.'P("') = 'P("')-'^'"^
IE! ,-eJ ieJ
do, (c la m o rpng cua (5 tren M°. Vay Y[ff, la M - khdp.
'*' • lel Gia sii (vii). Neu @N, la M - khop, thi theo (iii), Ni la M - khop vdi OP'
"^ , thAp Gia su F ngugc lai, Ni la M - khdp vdi mpi isl, chiing ta chiing minh 0 A f , l a M - ^ ^ '^
- fhi dd tp{F) la la modun con huu han sinh ciia M" va p : F > 0 N, la mot dong cau. t ^
modun con hiru han smh ciia @ /V,, hic la t6n tai tap con J hiru han ciia I sao c PV ) modun con cua A=QN.=Y[N,. Vi N, la M - khdp vdi moi isj, nen theo (vi), A la M - khdp. Do do ten tai d i n g ciu ip':F >A la m d rong ciia (p. Goi
S:A=^N. > 0 Af, la phep nhiing va dat i// = Sep . Chiing ta se chiing minh tf la md
rpng cua ip trenM". That vay, g p i / / : F >W la phep nhung, suy ra (p=<p' fi. Dieunay din din y//j{m) = S<p' tt(m) = Sgi(m) = tp(m),ytnsM". Chiing ta s u y r a yr la md rpng cua (p tren M°. Vay @ iV, la M - khdp.
TAI LIEU THAM KHAO
1. G. Connell (1963), On the group ring, Canad. J. of Math, 15, 650-685.
D. Farkas (1973), Self-injective group algebras,y. Algebra, 25, 313 - 315.
S. H. Mohamed and B. J. Muller (1990), Contitiuous attd Discrete Modules, London Math.
Soc. Lecture Note Series 147, Cambridge Univ. Press.
D. K. Tutuncu, R. Tribak (2010), On dual Baer modules, Glasgow Math. J., Vol. 52, No. 2, 2 6 1 - 2 6 9 .
Y. Shitov (2014), Group rings that the exact, J. Algebra, 403, 179 - 184.
D. Wilding, M. Johnson, M. Kambites (2013), Exact rings and semirings, J. Algebra 388, 324 - 337.
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