Modules With Chain Conditions On δ -Small Submodules

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1 Moduls With Chain Conditions On δ -mall ubmoduls Wasan Khalid Hasan, ahira Mahmood Yasn* Dpartmnt of Mathmatics, Collg of cinc, Univrsity of Baghdad, Baghdad, Iraq. bstract: Lt R b an associativ ring with idntity and M b unital non zro R-modul. submodul N of a modul M is calld a δ-small submodul of M (brifly N << M )if N+X=M for any propr submodul X of M with M/X singular, w hav X=M. In this work,w study th moduls which satisfis th ascnding chain condition (a. c. c.) and dscnding chain condition (d. c. c.) on this kind of submoduls.thn w gnraliz this conditions into th rings, in th last sction w gt sam rsults on δ- supplmnt submoduls and w discuss som of ths rsults on this typs of submoduls. Kywords: δ-small submodul, δ- supplmnt submoduls, c-singulr submodul. δ المقاسات التي تحقق خاصية السلسلة للمقاسات الجزئية الصغيرة وسن خالد حسن و ساهره محمود ياسين* قسم الرياضيات كلية العلوم جامعة بغداد بغداد الع ارق R حلقة تجميعية ذات عنصر محايد وليكن modul.n idal I of a ring R is δ-small idal if I is δ-small R-submodul of R. * sahira.mahmood@gmail.com M 218 الخالصة : لتكن ألمقاس الجزئيN منM يقال بأنه δ صغير اذا كان مقاسا احاديا غير صفري ايمن معرفا على R منM بحيثM/X كل مقاس جزئيX N+X=M منفردا فان X=M في هذا البحث سنقوم بد ارسة هذا النوع من المقاسات الجزئية والمقاسات التي تحقق خاصيتي السلسلة على المقاسات الجزئية δ صغيرة. كذالك قمنا بتعميم هذه الشروط على الحلقات وفي الجزء االخير حصلنا على بعض النتائج عن المقاسات الجزئية δ -المكملة وتوضيح بعض نتائجها. 1.Introduction Lt R b an associativ ring with idntity and M is a non zro unital right R-modul. submodul of R-modul is calld ssntial ( M)if vry non zro submodul of M has non intrsction with. M is calld singular modul if Z(M)=M whr Z(M)={x M:ann(x) R} submodul N of a modul M is calld a small submodul of M, dnotd by N << M, if N + L M for any propr submodul L of M [1]. In [2] Zhou introucd th dfinition of th concpt of δ-small submodul that a submodul N of a modul M is calld a δ-small submodul of M (brifly N << M )if N+X=M for any propr submodul X of M with M/X singular, w hav X=M. Lt M b an R- modul, a submodul X of M is calld c-singulr (X M) if is M/X singular

2 Rmark1-1[3] 1-lt b submodul of R-modul M if M thn C. M. 2-Lt M and N b R-moduls and f : M N b an pimorphism if C. M thn f() N. 3--Lt and B b submoduls of R-modulM if C. M and B C. M. thn ( B) C. M. 4- Evry submodul of a singulr modul is c-singulr. Lmma 1.2 [2]:Lt M b a modul, 1) For submoduls N,K, L of M with K N thn a) N << M if and only if K << M and N/K<< M/K b) N+L << M if and only if N<< M and L<< M. 2) If K<< M and f : M N a homo thn f(k) << N. 3) If K1 M1 M, K2 M2 M, and M=M1 M2 thn K1 K2<< M1 M2 if and only if K1<< M and K2 << M2. 4) Lt <B<M, If << M and B is a dirct summand thn << B. In [4], If N and L b submoduls of a modul M.N is calld a δ-supplmnt of L in M if M=N +L and N L<< M.and if vry submoduls of M. has a δ-supplmnt in M.Thn M is calld a δ-supplmnt modul n R-modul M is said to satisfy th ascnding chain condition (a.c.c.) on small submoduls. rspctivly dscnding chin condition (d.c.c.) on small submoduls if vry ascnding dscnding chain of small submoduls K 1 K 2 K 3. K n. rspctivly K 1 K 2.. K n Trminats[5]. In this work,w study th moduls which satisfis th ascnding chain condition (a. c. c.) and dscnding chain condition (d. c. c.) on δ-small submoduls.thn w gnraliz ths conditions into th rings. In th last sction w gt som rsults on δ- supplmnt submoduls and w discuss som of ths rsults on this typs of submoduls. 2.Moduls with chain conditions on a δ- small submoduls In this sction,w introduc th dfinition of modul which satisfis th ascnding chain condition (a. c. c.) and dscnding chain condition (d. c. c.) on δ-small submoduls as a gnralization of chain condition (a. c. c.) and dscnding chain condition (d. c. c.) on small submoduls [5] and w study th rlation btwn th ring that satisfis (a. c. c.) and dscnding chain condition (d. c. c.) on δ-small idals.. Dfinition (2.1): n R-modul M is said to b satisfis th ascnding chain condition (a.c.c.) on δ- small submoduls. rspctivly dscnding chin condition (d.c.c.) on δ-small submoduls if vry ascnding (dscnding) chain of δ-small submoduls K 1 K 2 K 3. K n. rspctivly K 1 K 2.. K n trminats. inc vry small submodul is δ- small submodul, Th following is clar Rmark (2.2):If M satisfy th a.c.c.(d.c.c.) on δ-small submoduls thn M satisfy th a.c.c.(d.c.c.) on small submoduls. Proposition (2.3):Lt M 1 and M 2 b two R-moduls and R=annM 1 +annm 2.Thn M 1 M 2 satisfis a.c.c.(d.c.c.) on δ-small submoduls iff M 1 and M 2 satisfis a.c.c.(d.c.c.) on δ- small submoduls. 219

3 Proof :inc R=annM 1 +annm 2, lt N 1 K 1 N 2 K 2 N 3 K 3. N n K n b ascnding chain on δ-small submoduls of M 1 M 2 hnc, N 1 N 2 N 3. N n is ascnding chain on δ- small submoduls of M 1 and K 1 K 2 K 3. K n b ascnding chain on δ-small submoduls of M 2. inc M 1 and M 2 satisfis a.c.c. on δ-small submoduls thn t, r Z such that N t = N =. i =1,2,3, and K r =K r i i =1,2,3, tak s=max{t,r },hnc N s + K s = N s i + K s i = i =1,2,3, Convrsly lt N 1 N 2 N 3. N n b ascnding chain on δ-small submoduls of M 1 thn N 1 {0} N 2 {0} N 3 {0}. N n {0} is an ascnding chain of δ-small submoduls of M 1 M 2, thn m Z such that N m {0}= N m i {0} i =1,2,3 thn N m = N m i i similar proof for d.c. c. Rcall that an R-modul M is calld multiplication if M=MI for som idal I of R.Th following proposition givs a rlation btn δ- small idals and δ- small submoduls of a a finitly gnratd faithful multiplication moduls. Proposition (2.4): Lt M b a finitly gnratd faithful multiplication R- modul, and lt N= M I, for som idal I of R thn N is δ-small submodul in M iff I is δ-small idal in R. Proof : ssum N is δ-small in M, and N = M I, lt I + J = R, for som c-singlr idal J of R. thn M I + M J = M R = M. thn M=N+MJ,sinc J is c-singulr idal in R thn J R[3,P.32] and by [6,prop.1.5] MJ M thn MJ C. M and sinc N is δ-small in M, thn MJ=M=RM thn J=R [7]. Convrsly, Lt N+K=M for som c-singlr submodul K of M,inc M multiplication R- modul, thn K=MJ, for som idal Jof R [6] Hnc N + K = M I + M J =M (I + J) = M But M is a finitly gnratd faithful multiplication R- modul, thn I + J = R, inc K=MJ singular modul. Lt thn (x+mj)l = MJ thn xl MJ If t i M I thn M/MJ x M/MJ, x MJ i. x is non zro thn x L = 0 for som L larg idal in R xl=0 thn L annm=0(m is faithful) thn L=0 which is a contraduction sinc L R thn xl 0. and xl xr 0., xl MJ xr MJ,hnc MJ M thn J R [6,Prop1.5.],thus J N is δ-small submodul in M. From th proof of Prop.2.4, w gt th following corollary. R[3,.p.32] thnj=r,j is δ-small idal in R. thus MJ=MR=M and thn Corollary (2.5).Lt M b a finitly gnratd faithful multiplication R- modul, and lt N= M I, for som idal I of R thn N C. M iff I C. R. Corollary (2.6) Lt M b a finitly gnratd faithful multiplication R-modul, thn R satisfis a. c. c. on c-singulr idal if and only if M satisfis a. c. c. on c-singulr submoduls. Proof: I 1 I 2 I 3.. I k.. b an ascnding chain of c-singulr idals in R thn by Corollary 2.5 M I 1 M I 2 M I 3. M I k... is an ascnding chain of c-singulr submoduls of M. inc M satisfis a. c. c. on c-singulr submoduls thn thn K N,, such that M I k = M I k+1 =. But M is a finitly gnratd faithful modul, thn I k = I k+1 =. k=1,2, Convrsly, Lt N 1 N 2 N 3.. N k.. b an ascnding chain of c-singulr submodul of M. inc M is a multiplication R-modul, thn N i = I i M, for som idal I i of R for all i. Hnc M I 1 M I 2 M I 3. MI k... But M is finitly gnratd thn by Corollary

4 I 1 I 2 I 3.. I k.. is an ascnding chain of c-singulr idals in R. inc R satisfis a.c.c on c-singulr idal, thn K N, such that I k = I k+1 =, hnc M I k = M I k+1 = which implis N k = N k+1 =, that is M satisfis a. c. c. on c-singulr submodul of M. Th following rsults ar squncs of this proposition. Corollary (2.7):Lt M b a finitly gnratd faithful multiplication R-modul, thn R satisfis a. c. c.( d.c. c. ) on δ-small idal if and only if M satisfis a. c. c.( d.c. c. ) on δ-small submoduls. Proof : Lt N 1 N 2 N 3.. N k.. b an ascnding chain of δ-small submodul of M. inc M is a multiplication R-modul, thn N i = I i M, for som idal I i of R for all i. Hnc M I 1 M I 2 M I 3.. M I k... But M is finitly gnratd thn by proposition (2.4) I 1 I 2 I 3.. I k.. is an ascnding chain of δ-small idals in R. inc R satisfis a.c.c on δ- small idal, thn K N, such that I k = I k+1 =, hnc M I k = M I k+1 = which implis N k = N k+1 =, that is M satisfis a. c. c. on δ-small submoduls. Convrsly, lt I 1 I 2 I 3.. I k.. b an ascnding chain of δ-small idals in R, thn by Proposition (2.4) M I 1 M I 2 M I 3. M I k... is an ascnding chain of δ-small submodul of M.inc M satisfis a. c. c. on δ-small submoduls thn thn K N,, such that M I k = M I k+1 =. But M is a finitly gnratd faithful modul thn I k = I k+1 =.. [7]. Thus R satisfis a. c. c. on δ-small idals of R. Proposition (2.8): Lt M b an R-modul, satisfis a. c. c. on δ-small submoduls.and is δ-small M submodul of M thn satisfis a. c. c. on δ-small submoduls of M Proof: Lt M..b a. c. c. on δ-small submoduls of thn 1 2 But is δ-small submodul and i M <<. thn i << M I [Lmma 1.2] thus is an ascnding chain of δ-small submodul of M. K N,, such that n = n 1 =..thus M satisfis a. c. c. on δ-small submoduls imilar proof for (d.c.c.).hnc w hav th following rsult : Thorm(2.9) : Lt M b a finitly gnratd faithful multiplication R-modul, thn th following ar quivalnt. 1) M satisfis a.c.c (d.c.c) on δ-small submoduls 2) R satisfis a.c.c (d.c.c) on δ-small idals. 3) =End R (M) satisfis a.c.c (d.c.c) on δ-small idals. 4) M satisfis a.c.c (d.c.c) on δ-small submoduls as - modul. Proof : (1) (2) By Cor (2.7) (2) (3)sinc M is a finitly gnratd faithful multiplication R-modul, thn R hnc R satisfis a.c.c(d.c.c) =End R (M) satisfis a.c.c (d.c.c) on δ-small idals. (3) (4) By Cor (2.7) 221

5 (4) (1) By Cor (2.7) R satisfis a.c.c (d.c.c) on δ-small idals. R [7] hnc R satisfis a.c.c(d.c.c) on δ-small idals and by cor (2.7) M satisfis a.c.c (d.c.c) on δ-small submoduls. 3.Moduls with chain conditions on δ- supplmnt submodul It is known that Rad(M) is th sum of all small submoduls of M. In [2]Zhou introducd th (M)as a gnralization of Rad(M). Dfinition 3.1 [2]: Lt b th class of all singular simpl moduls. For a modul M, Lt (M) = { N M, M/N }b th rjct M of. Lmma 3.2: [2,Lmma 1.5] Lt M and N b R- moduls 1) (M) = { L M / L is - small submodul of M } 2) If f :M N is an R-homomorphism thn f ( (N) ) (N). Thrfor (M) is a fully invariant submodul of M and M. (R R ) (M) 3) If M = i I M i, thn (M) = (M i ) 4)If vry propr submodul of M is containd in maximal submodul of M thn (M) is uniqu largst -small submodul of M. 5)Lt m M thn Rm<< M iff m (M). 6)n arbitrary sum of δ-small submoduls of M is δ-small submodul of M iff (M) << M. Rmark (3.3): Lt M b a finitly gnratd R-modul. Thn for any submodul of M, is δ- small iff (M). Proof :Clar from Lmma 3.2 and [1, Th ]. Proposition (3.4): Lt M b an R-modul thn th following ar quivlnt a)m satisfis a.c.c (d.c.c) on δ-small submoduls b)evry non mpty collction of -small submoduls posssss a maximal (minmal) mmbr. Proof : Clar. Proposition (3.5): Lt M b an R- modul thn M satisfis a.c.c on δ-small submoduls if and only if (M) is -small and vry δ-small submodul is finitly gnratd. Proof: ssum M satisfis a.c.c on δ-small submoduls Lt µ={b:b is a finit sum of δ-small submoduls of M } thn µ is non mpty collction of δ -small submoduls by[lmma 1.2] so by Prop.2.4 µ has maximal lmnt say K hnc K is δ-small submodul of M thn K (M). [Lmma 3.2,6].uppos that thr xists x (M). and x K hnc Rx is δ-small submodul of M [Lmma 3.2,5] so K+Rx is δ-small submodul thus K+Rx µ and K K+Rx this contraduction th maximality of K thn K= (M) thus (M). is δ-small submodul. Considr any δ-small submodul of M and lt G={B:B is finitly gnratd δ-small submodul of M,B M}sinc th zro submodul is containd in G,G,by Prop.3.4,G has a maximal lmnt say K,w claim that K=,inc K G,K is finitly gnratd and K. If K thn thr xist x,x K,hnc K+Rx is mmbr of G,contining K is contadaction maximality of K thn K= thn is finitly gnratd For th convrs,considr I 1 I 2 I 3.. I k.. an ascnding chain of δ-small submoduls,lt I= i I i thn I (M) sinc for vry i=1,2,3 I i (M).But (M) is -small submodul of M so I is -small submodul of M thus I is finitly gnratd I=Rx 1 +Rx Rx n now ach x i I i for vry i so thr xisit m such that x 1,x 2,.,x m I m,but this implis that I= I m so I m =I 1 m =.thus M satisfis a.c.c. on δ-small submoduls. From rmark (3.3) and similr proof of prop.(3.5) w gt th following Corollary (3.6): Lt R b aring thn R satisfis a.c.c on δ-small idal if and only if vry δ-small idal is finitly gnratd. Lt N and L b submoduls of a modul M. N is calld a supplmnt of L in M if M=N +L and 222

6 N L<<N.[8] In [4] If N and L b submoduls of a modul M. N is calld a δ-supplmnt of L in M if M=N +L and N L<< N.and if vry submoduls of M. has a δ-supplmnt in M.Thn M is calld a δ-supplmnt modul,r is calld a δ-supplmnt if it is supplmnt as R- modul.it is clar that vry supplmnt submodul is a δ-supplmnt but th convrs is not tru [4]. Proposition (3.7): ): Lt N and L b submoduls of a finitly gnratd faithful multiplication R- modul M such that N= M I, and L=NJ for som idals I,J of R thn N is δ- supplmnt submodul of L in M iff I is δ- supplmnt idal of J in R Proof :If N is δ- supplmnt submodul of L, thn M= N + L and N L<< M,thn M I + M J =M, MI MJ<< M hnc M(I+J) =M and M(I J)<< M thn R=I+J, by prop. 2.4 I J =<< I hnc I is δ- supplmnt idal of J in R as th sam proof th convrs is tru. Corollary (3.8): Lt M b a finitly gnratd faithful multiplication R-modul, thn R satisfis a. c. c(d.c.c.). on δ- supplmnt idal if and only if M satisfis a. c. c(d.c.c.). on δ- supplmnt submoduls. Proof : Lt I 1 I 2 I 3.. I k.. b an ascnding chain of δ- supplmnt idals of J i in R, thn M I 1 M I 2 M I 3. M I k... is an ascnding chain of δ- supplmnt submodul of J i M in M i=1,2,..by prop.3.7 thn K N,, such that M I k = M I k+1 =. But M is a finitly gnratd faithful modul, thn I k = I k+1 =.. [7]. Thus R satisfis a. c. c. on δ- supplmnt idals of R Convrsly, Lt N 1 N 2 N 3.. N k.. b an ascnding chain of δ- supplmnt submodul of L i i=1,2,.. inc M is a multiplication R-modul, thn N i = M I i and L i = M J i whr J i, I i idals of R for all i by prop. 4.1 I i ar δ- supplmnt idals of J i in R, hnc I 1 I 2 I 3.. I k.. is an ascnding chain of δ- supplmnt idals of J i in R. inc R satisfis a.c.c on δ- supplmnt idal, thn K N, such that I k = I k+1 =, hnc M I k = M I k+1 = which implis N k = N k+1 =, that is M satisfis a. c. c. on δ- supplmnt submoduls. Th sam argumnt for d.c.c. condition hnc omittd. Rfrncs 1. Kasch,F. 1982, Moduls and Rings, cadmic Prss Ins, London. 2. Zhou Y., 2000 Gnralizations of prfct, smiprfct and smirgulr rings, lgbra Coll., 7(3), pp: Coodarl K.R, 1976 Ring thory, pur and pplid Math., No.33 Marid-Dkkr. 4. Kosan M. T., lifting and -supplmntd Moduls, lgbra coll., 14(1),pp: Naoum.G. and Hadi I.M Modul with ascnding(dscnding) chain conditions on small submoduls, Iragi Journal of cinc, 37(3), pp: Naoum.G. and L-ubaidy W.K Moduls That satisfy a.c.c.(d.c.c.) on larg submoduls, Iraqi Journal of cinc, 45(1), pp: Naoum.G. 1984, on th ring of finitily gnratd multiplication moduls, Priodica Math. Hungarica,29, pp: Wisbaur R., Foundations of Modul and Ring Thory, Gordon and Brach, Rading. 223