V N U J o u m a l o f S c ie n c e , M a th e m a tic s - P h y sics 23 (2 0 ) 189 -1 Some results on (IEZ)-modules Le Van An1’*, Ngo Si Tung2 H ighschool o f Phan Boi Chau, Vinh City, N ghe An, Vietnam 2Department o f M athematics, Vinh University, N ghe An, Vìnam Received 16 A p r il 2007; received in rcvised ío rm 11 July 2007 A b s t r a c t A m odule A / is called Ự E Z ) —m odule i f fo r the subm odules i4, D , c o f M such that AC\ B = AC\C = B n C = 0, then A n (B ® C) = It is shovvn (1 ) Let any , j = 1, be u n iío rm local modules such that M i does not that: embed in J ( M j ) for Suppose that M = M \ ® M n is a ( / £ Z ) - m o d u l e Then (a) M satisfics (C ) (b ) The íoỉlovving assertions are cquivalent: (i) M satisfies (C ) (ii) I f X c (2 ) Let A / , (with i € then X c® M be u n ifo rm local modules such that A /t does not embed in J { Mj ) for any i , j = 1, ,71 Suppose that M = M \ © M n is a nonsing ular ( / E Z ) - m o d u le Then, M is a continuous module In tro d u ctio n Throughout this note, all rings arc associative with identity, and all modules are unital right mođules The Jacobson radical and the endmorphism ring o f M are denoted by J ( M ) and End(M) The notation X c e Y means that X is an cssential subm odulc o f Y For a module M consider the following conditions: (C i) Every submodule o f M is essential in a direct sum m and o f M (C ) Every submodule isomorphic to a direct sum m and o f M is itse lí a direct summand (C ) If A and D are direct sum m ands o f M with A n B = 0, then A ® B is a direct summand o f M A module M is dcfined to be a CS-module (or an extending m odule) if M satisíỉes the condition (C i) If M satisĩies (C i) and (C ), then M is said to be a continuous module M is called quasicontinuous if it satisíìcs (C i) and (C ) A module M is said to be a uniíbrm - extending if every uniform subm odule of M is essential in a direct summand o f M We have the following implications: Wc refer to [1] and [2] for background on c s and (quasi-)continuous modules In this paper, vve give some results on (7 E Z )-m o d u le s with conditions (C i), (C ), (C ) * Corrcsponding author Tcl.: 84-0383569442 E-mail: levanan_na@yahoo.com 189 190 L.v An, N.s Tung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 189-193 T he results A module M is called ( /f ? Z ) - m o d u le if for the submodules A, A n C = B n C = 0, then A n {B © C) = B ,c o f M such that ACì D E xam ples (a) Let F be a íĩeld We consider the ring F ^F ^ R = h Then Rfi is a Ự E Z ) —module Proof Let A, D , c be subm odules o f M — R r such that A n B = A n C — ũ n c = Then, there exist the subsets / , J, K o f {1, ,n } with I r\ J = I r\ K = J n K = such that M n Ẩ22 A = •• • 0 • • ^ \ 0 ::: A nn) where Aii = F Vĩ € I, and Aii = Vi e / ' , with / ' = { , (B n 22 0 b B = • Ko where Bu = F Vi J , and ^ = Vi e Bnn ị vvith J ' = (C n 0 c = C22 {1 • •• • ^ \ • 0 Cnn ị where Cii = F Vi € K , and Cu = Vi € K ' , with K ' = {1, ,n } \K Therefore, (xn B ®c = V 0 ũ \ X 22 ■■ 0 X where X u = F Vi ( J u K ), and X ii = Vi £ H , with H — { l , , n } \ ( J u K ) Ị n (J u ic) = 0, thus A n ( B © C ) = Hence Rft is a { I E Z ) ~ module Since L.v An, N.s Tung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 189-193 191 R e m a rk Let /F • 0 v° ! : ^ Mi = ( °0 0J o\ 0 Mn = v° ! Fì then Mi which are simple modules for any i = and R r = Mì © © M n vvhere R r in example Therefore, Mi are uniform local modules such that Mi does not embed in J [ M j ) for any í, J , n (b) Let F be a field and V is a vector space over fíeld F Set M = V © V Then M is not ( I E Z ) —module Proof Let /1 = { ( i , i ) I I Ễ V }, B = V ® 0, c = © V b e subm odules o f M We have A f ) B = A r \ C = B n c = bui A n { B ® C ) = A n M = A Hence, M is not ( I E Z ) ~ module We give two results on ( I E Z )-m o d u le with conditions (Ci), (C ), (C ) T h eo rem Let M i , M n be uniform local modules such thai Mị does nol embed in J ( Mj ) fo r any i , j — 1, Suppose thai M = M \ © © M n is ( I E Z ) —module Then (a) M satisfìes (C ) (b) The following assertions are equivalent: (i) M satisfies (Ứ2 ) (ii) I f X c M , X Sể Mi (with i e { , n}), then X c ® M T h eo rem Let M i , M n be uni/orm local moduỉes such thai Mi does not embed in J ( Mj ) fo r any i , j = , n Suppose ihal M = M \ © © M n is a nonsingular ( / E Z )-m odule Then M is a continuous module P ro o f of T heorem an d T heorem L cm m a ([3, L e m m a l.l]) Let N be a uniform local module such ihal N does not embed in J{N), íhen = E n d ( N ) is a local ring s L em m a Lel M \ , M n be uniform local modules such that Mi does not embed in J (Mj ) f or any Set M = A íi® ® A /n I f S \ , S c ® M; u - d i m ( S i ) = a n d u - d i m ( S ) = TI- 1, i,j = then M — S\ © i>2 Proof By Lemma we have E nd(M Ì) which is a Iocal ring for any i = By Azumaya’s Lemma (cf [4, 12.6, 12.7]), we have M = S ® K = S © M ị Suppose that i = 1, i.e., M = S2 đ Mi = â M \\ M = S\ â H = Sỡ đ (â je /M j) with I / 1= n - There are cases: 192 L.v An, N.s Tung / VNU Journal o f Science, Mathematics - Physics 23 (2007) 189-193 Case If Ệ I, then M = S\ © (M © © M n ) By modularity we get S\ © S '2 = ( 1S © S ) A / = ( i © 1S2 ) n ( © A /i) = S đ {{S\ â.S ) n M\ ) = S © í/, vvhere = ( 1S © S ) n Afj Therbre, u c A f |, = Sị = M ị By our assumption, w e must have u = M , and hence Si © S2 = £2 © Mi = M Case If I , then there is k Ỷ such that k = { l , , n } \ / By modularity we get S\ ® S = S © V, where V = (5 i © S ) n M \ Theròre, V c M i, V 2* S\ Mfc By our assumption, we must have V = M \, and hence i © S = © M \ = M , as desired Proof of Theorem (a ), We show that M satisĩies (C ), i.e., for tvvo direct summands S \ , S o f M with S \ n = 0, S i © S is also a direct summand o f M By Lemma we have End(Mi), i = 1, , n is a local ring By A zum aya’s Lemma (cf [4, 12.6, 12.7]), we have M = Sị(B H = S \ đ = (đ ie /M i)â (â ie /M i) (vvhere J = and M = S ® K = S © (© je E M j) = © ( © j e £ ^ j ) (where F = {1, , n } \ £ ) We imply S\ ^ and - © je F M j Suppose that F = {1, Let ự) be isomorphism @ị=lMj — ♦ S Set X j = we have = Ằfj, S = (Bị=ìXj By hypothesis S c ® M , we must have X j c ® A /, j = , k We show that S i ® = S\ ® (X i © © X k ) is a direct summand o f M We first prove a claim that S\ © X \ is a direct summand o f M By A zum aya’s Lcmma (cf [4, 12.6, 12.7]), we have M = X \ © L = X \ © (© sesA /s) = M a © (© sg sM a), with s c { , ,n } such that card(S) = n — and a = { l , , n } \ Note that card(S n / ) > c a r d ự ) — — m Suppose that {1, c ( n / ) , i.e., M = (5 i © (M j © © M m )) © M/J = z © M[) with = /\{ , and z = i © (M i © © M m) By M is a ( / £ Z ) - m o d u l c and X i n i = X i n (Mị © © M,n) S\ n (M i â đ M m) = 0, we have z n X \ = By z , X i c ® M , u — dim(Z) = n — 1, u — dim(X ) = 1, i.e., u — dim(Z) + u — dÌTn(X\) = TI and by L em m a we have M — z ® X i = Si © (A /j © © Mrrì) © — {S\ © ->^1 ) © (A/ị © © M m) Therefore, Si ® Xỵ c ® M By induction vve have Si © = S\ © ( Xi © © Xk) = (S\ © X \ © © X k - \ ) © Xk is a direct summand o f M , as desired (ò), T he implication (i) = > (ii) is clear (n ) = > (i) We show that M satisfies (C ), i.e., for two subm odules X , Y o f M , with X ^ Y and Y c ® M , X is also a direct summand o f M Note that, since u — d i m ( M ) = TI, we have u — dim {Y ) — , , n, the following case is trival: u — d im ( Y ) = If u - d i m ( Y ) = , ,71 By Azum aya’s Lemma (cf [4, 12.6, 12.7]) X — Y = © i g /M i ,/ c { l , ,n } Let A n Mj Ỷ 0- By property A and Mj are uniform L V An, N.s Tung / VNU Journal o f Science, Maihematics - Physics 23 (2007) 189-193 193 submodules we have A n M j c e A and A n M j c e M j By A and M j are closure o f A n M j, M is a nonsingular module, we have A = Mj c ® M This implies that M is uniíòrm - extending Since M has íìnite uniíorm dim ension and by [1, Corollary 7.8], M is extending m odule, as desired Proof ofTheorem By Lemma 3, M is a C S - module We show that M satisĩies (C ) By Theorem , we prove that if X c M y X — Mị (with i E then X c M Sct X is a closure o f X in M Since Mi is a uniíòrm module, thus X is also uniíbrm Therore X * is a uniíbrm closed module We imply X * is a direct summand o f M We have X * = M j, thus X c Afj If X c M j , x Ỷ M j then X c Hence Mị = X c a contradiction We have X = Mj c e A f, as desired Acknovvlcdgm ents The authors are gratef\il to Prof Dinh Van Huynh (D epartm ent o f M athematics Ohio U niversity) for many helpful comments and suggestions The author also w ishes to thank an anonymous referee for his or her suggestions which lead to substantial improvements o f this paper R eíerenccs [1] N V Dung, D.v Huynh, p F Smith, R Wisbaucr, Extending Modules, Pitman, London, 1994 [2] S.H Mohamed, BJ Muller, Continuous and Discrete Kíodules London Math Soc Lecture Notc Ser Cambridge University Press, Vol 147 (1990) [3] H.Q Dinh, D.v Huynh, Somc Results on Self-injective Rings and E-CS Rings, Comm Aỉgebra 31 (2003) 6063 [4] F.w Anderson, K R Furlcr, Ring and Categories o/Modules , springer - Verlag, NcwYork - Hcidelberg - Berlin, 1974 [5] K.R Goodcarl, R.B Warfield, An ỉntroduction to Noncommutative Noeíherian Rings, London Math Soc Student Text, Cambridge Univ Press, Vol 16 (1989) [ ] D V Huynh, s K Jain, s R Lỡpez-Permouỉh, Rings Characterized by Direct Sum o f CS-modules, Comm Algebra 28 (2000) 4219 [7] N.s Tung, L.v An, T.D Phong, Somc Rcsults on Direct Sums o f Uniform Mcxiules, Contributions in Math and Applications, ICM A, December 2005, Mahidol ưni., Bangkok, Thailan, 235 [ ] L.v An, Somc Rcsults on ưniĩonn Local Modules, Submitted ... T.D Phong, Somc Rcsults on Direct Sums o f Uniform Mcxiules, Contributions in Math and Applications, ICM A, December 2005, Mahidol ưni., Bangkok, Thailan, 235 [ ] L.v An, Somc Rcsults on ưniĩonn... Pitman, London, 1994 [2] S.H Mohamed, BJ Muller, Continuous and Discrete Kíodules London Math Soc Lecture Notc Ser Cambridge University Press, Vol 147 (1990) [3] H.Q Dinh, D.v Huynh, Somc Results on. .. Anderson, K R Furlcr, Ring and Categories o/Modules , springer - Verlag, NcwYork - Hcidelberg - Berlin, 1974 [5] K.R Goodcarl, R.B Warfield, An ỉntroduction to Noncommutative Noeíherian Rings, London