VNU JOURNAL OF SCIENCE M athem atics - Physics T X V III N q2 - 2002 E X T E N D IN G PRO PERTY O F IN F IN IT E D IR E C T S U M S O F U N IF O R M M ODULES N go Si T u n g D e p a rtm e n t o f M a th e m a tic s, V inh U n iversity B u i N h u Lac D e p a rtm e n t o f M a th e m a tic s, N am D inh C ollege o f P ed a g o g y I n tr o d u c tio n Let R be any ring and M = © ,e/A /i be a direct sum of uniform right R - submodules M i ,i £ / We are interested ill the question, when this module M is extendible If the index set / is finite, this question has been studied by Harmanci and Sm ith Ị6) In deed, Harmanci and Sm ith have shown th at if M = 0™=1Aft, then M is extendible if and only if every direct summand of A/ with uniform dimension is an extending module, where each M t is uniform, [(), Theorem 3] In the first part of this paper we give some conditions for A/ to be extendible, where the index set / is not necessarily finite We show th a t a module over ail arbitrary ring is extendible if it has (1 —C l) and every local direct sum m and is a direct summand Moreover, properties of extending modules have been obtained In the last part, of the paper the results have been applied to characterize quasiFrobeniüS rings and rings whose projective right modules are extendible P re lim in a rie s Throughout this paper all rings R are associative rings with identity and all R modules are unitary right R- modules We consider the following conditions on a module A/: (Cl) Every submodule of M is essential in a direct sum m and of M (C ) Every submodule isomorphic to a direct sum m and of M is itself a direct summand of M (1 - C l) Every uniform submodule of M is essential in a direct sum m and of A/ A module M is called co n tin u o u s if it satisfies conditions (C l) and (O ), V -co n tin u o u s if it satisfies (C ) and (1 —C l) A module M is said to be an extending m odule if it satisfies condition (CY), and M is said to have the exten d in g property o f u n ifo rm subm odule if it satisfies condition (1 -Cx) A submodule y\ of a module A Í is closed in M , it it has no proper essential extensions in M It is easy to check th a t ' M is an extending module if and only if every closed submodule of M is a direct sum m and of M 52 E x t e n d i n g p r o p e r ty o f in fin ite direct s u m s o f u n i f o r m m o d u le s 53 A moduli’ M is s.lid to have finite uniform dimension if M does not contain an infinite direct Slim non-’/rrn sn!>niodul(\ A ring IỈ is Qiiiisi - Frobc?nii.is (brioilv Q F ) if ÍỈ is right artinian and right self - ilijfTtivn It is knn\v that a ring IỈ is Q F if every projectiw /? - moduli' is injrct ivf* if Í*very injcr.l ivr IỈ - m o d u lo is projectiVC? (See [4, T h e o r e m 24.20]) A ring f i is E —> w h ere E is injcTt.ivr a n d korf is sm all in Py p is ilijcH't iv/*.- injective For this j)\irpose let be a sunmodule of Mjỗ and a be a homomorphism N q o S i T u n g , B u i N h u Lac 56 of u in M (J) We show th a t a is extended to one in Hom ft (Aí*, M (./)) Since M ( J ) ® Mk has (1 —C l), there is a direct sum m and A"* of A/ such th a t { x -a (æ ) : x € Ư} c e x \ Since as a direct sum m and of A/, M (J)© M fc has a decom position th at complements direct summands We consider two cases: a) M (J ) © Mk = X * © M ( J f), where J r is a subset ofT hen M ( J ; Mk = À'" © M ( J ) ầ A â A /(J) ầ A /(J) â Mfc Hence I * e M (./') = X* © A /(./) It follows that J = J Therefore nIA/fc extends a where n : X *© A/(.7) —> M ( J ) is the projection b) A /(J) © M k = X* © M ( J \ ) © Mk , where J] is a subset of ,/ Let rifc : X m © M ( J \ ) © Mk -» A//c be the projection and let = ( X * © M( J i ) ) n M Ụ ) If A Ỷ and suppose th a t A n M j — Í for each j -J= J , then by [3 Proposition 3.6], A is essential in M ( J ) Then it is easy to check th a t X * © A is essential in A/a © M ( J ) Hence A/fc n ( X * © M (J )) Ỷ 0» a contradiction Consequently there exists j € J such that M j n A = Hence M j n k e rn * = and thus M j — ĩ ỉ k { Mj ) By hypothesis we have n*(A /j) = A h , Therefore we have A'* © M ( J \ ) © M k = X* © A/(.A) © M j = A'* e M (J i) where /2 = /1 u {/} Hence we may use a) to show that a is extxmđeđ to one in hom *(A 4, M (J )) Now assume th at = Then M ( J \ ) = It implies th at A /(./)© A/a = X*©A/fc From this, it is easy to see th at M ( J ) is uniform and M ( J ) © M k = Afj © A4 = X * © A/*, where J = {j} is a set of only one element Hence we have IljJA'/j) = A/it and Afj © A/fc = X * ® M There fore II extends Q, where n : X * -> M j is the projection, proving (iii) (iii) =» (t) Let /4 be a closed subm odule of A'/, and be a subset of I which is maximal with respect to A n M ( J ) = Then it is easy to see th at /1 M(.7) is essential in M Let K = \ \ J and T i f t , l \ j be the projections of M onto M ( K ) and A /(J), respectively T hen H/ (i) by Theorem 14 ( i n ) => (?) Since R is left perfect, each e rR contains a uniform submodule, where {r,}ỊỊ_i is a set of orthogonal primitive idem potents of R Since each e t R has (1 —C l), CiR is uniform for all < i < n Then every projective right R -module p has a direct sum decomposition p = i €/P ii where each Pi is uniform As in the proof of Theorem 14 we see that, p is an extending module By [9, Theorem II], R is a right CO‘H ring For Q F - rings we prove the following T h e o re m 16 For a ring R , th e follow ing s ta te m e n ts are equivalent: (i) R is Q F (ii) R is r ig h t p e r fe c t , right co n tin u o u s a n d th e p ro je c tiv e cover o f e v e r y sem isim p le R -m o d u le h a s (1 - C l ) N qo S i T u n g , B u i N h u Lac 58 (in) R is right CO — // a n d rig h t continuous ( i v ) R is rig h t CO — /7 a n d r ig h t u - c o n t in u o u s (v) R is a right H -rin g a n d right c o n tin u o u s (vi) /? is a right li- r in g a n d rig h t Ư -continuous Proof Since /Í is right continuous, Z { R f i ) = by [11, Lemma 4.1] Hence, by |9, Theorem 4.3] we have (t) (Hi) and (i) (v) (z) => ( i i ) y ( n i ) => ( iv ) and ( v ) =$■ ( v i ) are clear («) => (in ) Since /? is right perfect and right continuous, R n has finite uniform dimension By Theorem 14 R is a right CO —H ring (iv) =$■ ( i n) By Corollary 15, R is right perfect Hence R is right continuous, by Corollary 11 (m) => (v) By 9, Theorem 2.11], R is right artinian Hence, ÍỈ is right, continuous by Corollary 11 A c k n o w le d g m e n t: We would like to thank Professor Dinh Van Huynh very much for calling our attention to the study of extending modules and for many helpful discussions R e fe re n c e s F w Anderson and K.R.Fuller, R in g s and C ategories o f M odules, Springer - Verlag 1974 A w C hatters and C.R Hajarnavis, Rings in which every com plem ent right ideal is a direct sum m and, Q uai J M ath O xford (2 ) 28(1977), 61 - 80 Ng V Dung - D V Huynh - p F Smith and R W isbauer, E xten d in g M odules , Pitm an London, 1994 c G Faith Algebra II R ing Theory ; New York, Springer - Verlag, 1976 M Harada, Oil modules with extending property O saka J M a th , 19(1982), 203 215 A Harmarici and p F Sm ith, Finite direct sum s of c s - modules, H ouston M ath., 19(1993), 523 - 532 M A Kamal and B J Millier, T he structure of extending modules over noetherian rings, O saka J M a th , 2Ồ(1988), 539 - 551 H Mohamed and J Muller, C o n tin u o u s and D iscrete M odules, L o n d o n Math Soc Lecture Note series 147, Cambridge Univ Press, Cambridge, 1990 K Oshiro, Lifting modules, extending modules and their applications to QF- rings, H okaido M ath J., 13(1984), 310 - 338 10 Phan Dan, Right perfect, rings with the extending property oil finitely generated free modules, O saka J M ath., 26(1989), 265 - 273 11 Y Utumi, On continuous rings and self - injective rings, Trans A m e r M ath Soc 118(1965), 158 - 173 12 N Vanaja and Vandana M Purav, C haracterizations of generalized uniserial rings, C o m m u n ica tio n s in Algebra, 20 (8) (1992), 2253 - 2270 E x te n d in g p ro p e rty o f in fin ite d ire c t, s u m s o f u n ifo r m m o d u le s TAP CHÍ KHOA HỌC ĐHQGHN Toán - Lý T.XVIII, Số - 2002 TÍNH CHẤT MỞ RỘNG CỦA T ổ N G T R Ự : TIẾP VÔ HẠN CỦA CÁC M ÔĐUN ĐỀU Ngơ SI Tùng Khoa Tốn, Dại học Sư phạm Vinh Bùi Như L c Trường Cao dẳng Sư phạm Nam Định Cho M = © ,e;A /?, M i mơđun đểu I tập vô hạn bất kỳ, câu hỏi đạt M c s - môđun Trong báo đưa số điéu kiện đế môđun M c s thông qua lớp (1 - Ci)-m ỏđun Các kết thu mờ rộng số kết A Kamal J.Millier [7] Phan Dan [10] ... odule of A/ By Corol lary , A contains a uniform direct sum m and X of M , Hence we can define a non-em pty set V of direct sums of uniform modules in M as follows: V = {âcvÊA^ ầ Ay Aft is uniform. .. contains a uniform direct sum m and B p of M T hen A © B(i is member of V , a contradiction to the rnaximality of A ' Hence B = 0, i e A — A ' is a direct summand of M Thus M is an extending. .. - C l) Proof Let M be an extending module Obviously, M has (1 —C l) Conversely, let M ■ y ]í(?J M ị where each M l is uniform and assume th at any local direct sum m and of M is a direct sum