IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Essentially Quasi-Invertible Submodules and Essentially Quasi-Dedekind Modules I.M-A Hadi , Th. Y. Ghawi Departme nt of Mathematics , College of Education I bn AL-Haitham Unive rsity of Baghdad Departme nt of Mathematics , College of Education, Unive rsity of AL- Qadisiya Received in : 6 June 2011 Accepte d in : 8 February 2011 Abstract Let R be a commutative ring with identity . In this p aper we st udy the concepts of essentially quasi-invertible submodules and essentially quasi-Dedekind modules as a generalization of quasi-invertible submodules and quasi-Dedekind modules . Among the results that we obtain is the following : M is an essentially quasi-Dedekind module if and only if M is aK-nonsingular module,where a module M is K-nonsingular if, for each )(MEndf R  , Kerf ≤e M imp lies f = 0 . Kew words : Essentially quasi-invertible submodules , Essentially quasi-Dedekind M odules . Introduction The concepts of a quasi-invertible submodule of an R-module and quasi-Dedekind module were introduced in [5] .Where a submodule N of an R-module M is called quasi- invertible if 0),( MNMHom , and an R-module M is called quasi-Dedekind if each nonzero submodule of M is quasi-invertible . As a generalizations to these concepts we introduce the following concepts : We call a submodule N of M is essentially quasi- invertible if , N ≤e M and N is quasi-invertible .And an R-module M is called essentially quasi-Dedekind if every essential submodule N of M is quasi-invertible ; ( i.e 0),( MNMHom ) . This p aper consist s of t wo sections , §1 is devoted to study essentially quasi-invertible submodules , in §2 we st udy and give the basic p rop erties of essentially quasi-Dedekind modules . This p aper represents a p art of the M . Sc. thesis written by the second author under the sup ervision of the first author and was submitt ed to the college of education – Ibn AL- Haitham , University of Baghdad , 2010 . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 1 . Essenti ally Quasi-Invertible Submodul es In this section we introduce the concept of essentially quasi-invertible submodules. We develop basic p rop erties of essentially quasi-invertible submodule . We st art with the following definition : De finition (1.1) Let M be an R-module and N ≤e M , then N is called an essentially quasi-invertible submodule of M if , 0),( MNMHom ; that is N is essentially quasi-invertible if , N ≤e M and N is quasi-invertible . An ideal J in a ring R is called an essentially quasi-invertible ideal of R if , J is an essentially quasi-invertible R-submodule of R . Remarks and Exam ples (1.2) 1) It is clear that every essentially quasi-invertible submodule is quasi-invertible submodule . Recall that an R-module M is called a semisimple if every submodule of M is a direct summand of M , [3, p .189] . 2) If M is a semisimp le R-module , then M is the only essentially quasi-invertible submodu le of M . 3) Consider Z4 as a Z-module , )2(N ≤e Z4 , but 0)),2(( 244  ZZZHom , so )2(N is not essentially quasi-invertible submodule of Z4 , similarly in the Z-module Z20 , )2(N ≤e Z20 , but it is not quasi-invertible . 4) If N is an essentially quasi-invertible R-submodule of an R-module M , then NannMann RR  . Proof : It is clear . The converse of (Rem.and.Ex. 1.2(4) ) is not true in general , for examp le : Let ZZM  , considered as a Z-module and let MZN  )0( , then it is clear that NannMann RR  = (0) , but N is not essentially quasi-invertible submodule of M , since N ≰e M and also N is not quasi-invertible . 5) Let J be an ideal of a ring R . Then J is an essentially quasi- invertible if and only if 0)( JannR . Proof : It is easy . 6) Let J be an ideal of a rin g R . The followin g st atements are equivalent : a) J is an essentially quasi- invertible ideal of R . b) J is a quasi-invertible ideal of R . c) 0)( Jann R . Proof : )()( ca  : It follows by ( Rem.and.Ex. 1.2(5) ) . )()( cb  : It follows by [5 , p rop . 2.2] . 7) Let R be a ring . The following st atements are equivalent : IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 a) R is an inte gral do main . b) R is quasi-Dedekind . Proof : It follows by (Rem.and.Ex. 1.2(6)) . 8) If 21 MMM  is an R-module , and K be an essentially quasi- invertible submodule in M i for some i= 1,2 , then it is not necessarily that K is an essentially quasi-invertible submodule of M , for examp le : Let 2ZZM  as Z-module , then K = Z2 is an essentially quasi- invertible submodule of Z2 as Z-module , but 22 )0( ZZ  which is not essentially quasi-invertible of 2ZZM  , since 2)0( Z ≰e 2ZZ  . Proposi tion (1.3) Let M be an R-module , and let N1 , N2 be an essentially quasi- invertible R- submodules of M , then 21 NN  is an essentially quasi-invertible R-submodule of M . Proof : Since N1 ≤e M , N2 ≤e M then 0),( 1 MNMHom and 0),( 2 MNMHom . Also N1 ≤e M , N2≤e M imp ly 21 NN  ≤e M . But ),(),(),( 2121 MNMHomMNMHomMNNMHom  .Hence 0),( 21  MNNMHom and so that 21 NN  is an essentially quasi- invertible R- submodule of M . The following lemma is needed for the next p rop osition . Lemma (1.4 ) Let M be an R-module such that for each nonzero submodule K of M , PPp MK 0 for each maximal ideal P of R . If NP ≤e M p imp lies N ≤e M . Proof : Sup p ose that there exists MU 0 such that 0 NU .Hence PPNU 0)(  which implies that PPP NU 0 , but PPp MU 0 by hy p othesis , so t hat NP ≰e M p which is a contradiction . Proposi tion (1.5) Let M be an R-module , N ≤ M . If NP is an essentially quasi-invertible RP - submodule of RP-module M P (for each maximal ideal P of R ) , then N is an essentially quasi-invertible submodule of an R-module M . Proof : Since NP is an essentially quasi-invertible RP-submodule of M P , 0),( PPP MNMHom . But by [4 , Ex.3 , p .75] , 0),()),((  PPPP MNMHomMNMHom , thus 0)),(( PMNMHom and by [4, Prop .3.13 , p .70] , 0),( MNMHom ; that is N is a quasi-invertible submodule of M . Beside this , by (Lemma (1.4 )) , N ≤e M . Thus N is an essentially quasi-invertible submodu le of M . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Recall that an R-submodule N of an R-module M is called a SQI-submodule if , for each ),( MNMHomf  , f(M /N) is a small submodule in M , [6 , p .44] . And an R-submodule N of an R-module M is called a small submodule of M ( N ≪ M , for short ) if , for all K ≤ M with N+K = M imp lies K = M , [3, P.106] . Remark (1.6 ) It is clear that every quasi-invertible submodule is an SQI-submodule and hence every essentially quasi-invertible submodule is an SQI-submodule . The converse of (Remark 1.6 ) is not true in gener al , consider the followin g example . Exam ple (1.7 ) Consider the Z-module Z4 , )2(N , then N is an SQI-submodule of Z4 , since for all )),2(( 44 ZZHomf  , then )2(( 4Zf ≨ Z4 , and every p rop er submodule of Z4 is a small in Z4 , so )2(( 4Zf ≪ Z4 , but it is known that )2(N is not essentially quasi- invertible in Z4 ,( see Rem.and.Ex. 1.2(3)) . 2 . Essenti ally Quasi-Dedekind Modul es In this section we give the definition of essentially quasi-Dedekind module with some examp les . We p rove that essentially quasi-Dedekind module and K-nonsingular module which is introduced by [8] are equivalent .We give conditions under which submodule (resp . quotient module) of essentially quasi-Dedekind is essentially quasi- Dedekind . De finition (2.1) An R-module M is called essentially quasi-Dedekind if , 0),( MNMHom for all N ≤e M . A ring R is essentially quasi-Dedekind if R is an essentially quasi- Dedekind R-module . Remarks and Exam ples (2.2) 1) It is clear that every quasi-Dedekind module is an essentially quasi- Dedekind module, but t he converse is not true in general , for examp le : Each of Z10 , Z15 are essentially quasi-Dedekind as a Z-module , but it is not quasi-Dedekind . 2) Every integral do main R is an essentially quasi-Dedekind R-module, by [5 ,Ex 1.4 , p .24] and (Rem.and.Ex 2.2(1)) . 3) Z4 as a Z-module is not essentially quasi-Dedekind , since )2( ≤ e Z4 , but 0)),2(( 244  ZZZHom . 4) Let M = Z p ∞ as a Z-module . Then M is not essentially quasi- Dedekind , but )(MEndZ ( is the ring of P-ad ic inte gers) is a commutative domain [see Ex 4.1.2 ,8] , so )(MEndZ is essentially quasi-Dedekind , by (Rem.and.Ex 2.2(2)) . 5) Let M be a uniform R-module . Then M is a quasi-Dedekind R-module if and only if M is an essentially quasi-Dedekind R-module . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Proof : It is clear . Roman C.S in [8] , introduce the following : " An R-module M is called K-nonsingular if , for each )(MEndf R , Kerf ≤ e M imp lies f = 0 " . However we prove the following : The orem (2.3 ) Let M be an R-module . Then M is an essentially quasi-Dedekind R-module if and only if M is a K-nonsingular R-module . Proof : ) Let )(MEndf R , 0f . Sup p ose that Kerf ≤ e M , defined MKerfMg : by g (m+Kerf) = f (m) for all Mm  . It is easy to see that g is well-defin ed and g is a nonzero homomorphism . Thus 0),( MKerfMHom which is a contradiction , since M is an essentially quasi-Dedekind R-module . ) N ≤ e M . Sup p ose that there exists MNMf : and 0f . we have MNMM f   ,where π is the canonical p rojection .Let )(MEndfo R  . KerN  and N≤ eM imp lies Ker ≤eM , ( ) ( ) ( ) 0M fo M f M N    which is a contradiction with M is a K-nonsingular R-module . Although the concepts of essentially quasi-Dedekind module and K-nonsingular module are equivalent ,but we see that it is convenient to use the notion essentially quasi- Dedekind in this p aper . Proposi tion (2.4 ) Every semisimple R-module is an essentially quasi-Dedekind R-module. Proof : It is easy . The converse of (Prop 2.4) is not true in general, consider the following examp le . Exam ple (2.5 ) It is known that Z as a Z-module is essentially quasi-Dedekind , but it is not semisimple . Recall that an ideal I of a ring R is semip rime if , for all Rr  with Ir 2 imp lies Ir  [or , for all ideal A of R with IA  2 imp lies IA  ] .And a ring R is called semip rime if (0) is a semip rime ideal of R ; i.e R does not contain nonzero nilp otent ideals , [2] . Proposi tion (2.6 ) Let R be a ring . The following st atements are equivalent : 1) R is an essentially quasi-Dedekind rin g . 2) R is a semiprime rin g . 3) Z(R) = 0 ( R is a nonsingular ring ) . Proof : )3()2(  : It is follows by [2 , Prop 1.27, p .35] )1()2(  : Let )(REndf R such that Kerf ≤ e R . To p rove f = 0 . Sup p ose that 0f , there exists Rr 0 such that f(a) = ra for all Ra  . Since Kerf ≤e R and Rr 0 , then there exists Rt 0 such that Kerfrt 0 , hence 0 = f(rt) = rf(t) = r 2 t . This imp lies (rt) 2 = 0 and since R is semip rime , rt = 0 which is a contradiction . Thus f = 0 and R is essentially quasi-Dedekind . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 )3()1(  : Sup p ose that 0)( RZ . Then there exists )(0 RZa  and hence )(aann R ≤ e R , this imp lies )(aann R is a quasi-invertible ideal and so t hat by (5 , Prop 2.2) , 0))(( aannann RR , but ))(()( aannanna RR , hence a = 0 which is a contradiction . Proposi tion (2.7 ) Let R be a ring . Then R is essentially quasi-Dedekind if and only if R[x] is essentially quasi-Dedekind , where R[x] is the ring of p olynomials with one indeterminate x . Proof : ) Sup p ose that R is essentially quasi-Dedekind , so by (Prop 2.6) R is a nonsingular rin g , and hence by [2 , Ex. 13, p .37] , R[x] is a nonsingular rin g . Thus R[x] is essentially quasi-Dedekind , by ( Prop 2.6) . ) Sup p ose that R is not essentially quasi-Dedekind , so by (Prop 2.6) , R is not a semip rime rin g ; that is there exists )(RLa  and oa  , where 0:{)(  n xRxRL , for some }Nn  , then an = 0 , for some n N . Define 0)(  axf , so ][)( xRxf  , and R[x] is a semip rime rin g , by (Prop 2.6) . On the other hand [ f(x) ] n = a n = 0 , implies 0])[()(  XRLxf . It follows that f = 0 which is a contradiction . Thus R is essentially quasi-Dedekind . Proposi tion (2.8 ) Let M be a faithful R-module . Then R is essentially quasi- Dedekind if and only if N M N  is a faithful R-module , for all MN  . Proof : ) Sup p ose that R is essentially quasi-Dedekind , so by ((Prop 2.6), R is semip rime . Let )( N M Nannr R  , then )()( N M annNannr RR  ; that is rN = 0 and NrM  , so 0 2  rNMr imp lies 0)(2  Mannr R then 0 2 r , thus r = 0 , since R is a semip rime rin g . Therefore N M N  is a faithful R-module for all MN  . ) Sup p ose that N M N  is a faithful R-module , for all MN  . To p rove that R is essentially quasi- Dedekind . We shall p rove that R is a semiprime ring . Let Rr  with 02 r , sup p ose that 0r , so )(Mannr R , since M is a faithful R-module, then 0rM .Let MrMN  , hence rN = r2M = 0 , so )(Nannr R , but )( N M annr R ( since NrMrM  ) , so 0)()()(  N M Nann N M annNannr RRR , thus r = 0 which is a contradiction. Hence R is essentially quasi-Dedekind . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Proposi tion (2.9 ) Let M be an R-module and let JRR  , where J is an ideal of R such that )(MannJ R . Then M is an essentially quasi-Dedekind R-module if and only if M is an essentially quasi-Dedekind R -module. Proof : By [3, p .51] , we have ),(),( MNMHomMNMHom RR  for all MN  . Sup p ose that M is an essentially quasi-Dedekind R-module , then 0),(),(  MNMHomMNMHom RR for all N ≤ e M , imp lies M is an essentially quasi-Dedekind R -module . The converse follows similarly . Let R be an integral domain , and let M be an R-module . An element Mx  is called a torsion element of M if , 0)( xann R . The set of all torsion elements of M denoted by T(M ) and it is a submodule of M . If T(M ) = 0 the R-module M is said to be torsion-free , [1, p.45] . The following result shows that essentially quasi-Dedekind p reserves under isomorphism . Proposi tion (2.10 ) Let M 1 , M 2 be R-modules such that 21 MM  . Then M 1 is an essentially quasi- Dedekind R-module if and only if M 2 is an essentially quasi-Dedekind R-module . Proof : ) Sup p ose that M 1 is an essentially quasi-Dedekind R-module . Let 21 : MM  ,  is an isomorphism . To p rove that M 2 is an essentially quasi- Dedekind R-module . Let 0,)( 2  fMEndf R . We have 1221 1 MMMM f    , let )( 1 1 MEndofoh R    , and hence 0h , then Kerh ≰e M 1 . To p rove Kerf ≰e M 2 , we cliam that })(:{ 1 2 KerhyMyKerf   , to p rove our a sseration . Let 0)(,  yfKerfy ,   ))(())()(())(( 1111 yofyofoyh  0)0())(( 11    yf .Then for all Kerfy  , Kerhy   )(1 , so KerhKerf  )(1 ≰e M 1 which imp lies )( 1 Kerf  ≰e M 1 , so Kerf ≰e M 2 . Thus M 2 is an essentially quasi-Dedekind R-module . ) The proof is similarly . Remark (2.11 ) Let M be an R-module and let MN  . If NM is an essentially quasi- Dedekind R-module . Then M is not necessarily an essentially quasi-Dedekind R-module , as we can see by the following examp le . Exam ple (2.12) Let M = Z 4 as a Z-module , and )2(N ≤ Z4 , then 24 )2( ZZ  is an essentially quasi-Dedekind Z-module ,but M = Z 4 is not an essentially quasi-Dedekind Z-module . Now, we turn our att ention to a submodule of essentially quasi-Dedekind. First consider the following remark : IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Remark (2.13 ) Let M be an essentially quasi- Dedekind R-module , MN  . Then it is not necessarily that N be an essentially quasi-Dedekind R-module .To show this , consider the following examp le which ap p eared in [7] . Let 2ZQM  as a Z-module is essentially quasi-Dedekind . Take 22 ZQZZN  as a Z-module , then N is not essentially quasi- Dedekind as a Z-module , since if NNf : define by ),0(),( xyxf  , 2, ZyZx  , then 0f and 2 2}0:),{()}0,0(),(:),{( ZZxNyxyxfNyxKerf  . Hence Kerf ≤ e N . Thus 2ZZN  is not an essentially quasi-Dedekind as a Z-module. Now , in the next p rop osition we give a condition which makes R-submodule of an essentially quasi-Dedekind R-module is essentially quasi-Dedekind . Proposi tion (2.14 ) Let M be an essentially quasi-Dedekind R-module , and M is quasi – injective. If N ≤ e M t hen N is an essentially quasi-Dedekind R-module. Proof : Let )(NEndf R , 0f , to p rove that Kerf ≰e N . Assume that Kerf ≤ e N . Since M is quasi–injective , then there exists )(MEndg R such that goi = iof , ( where i is the inclusion map p ing) . It follows that 0g , and this imp lies Kerg ≰e M , since M is essentially quasi- Dedekind . But KergKerf  , so Kerf ≰e M . On the other hand N ≤ e M and by assump tion Kerf ≤ e N imp ly Kerf ≤ e M . To show this , since N ≤ e M then for all MU  , 0U then 0UN and NUN  .But Kerf ≤ e N , hence 0)(  UNKerf ; that is 0)(  NUKerf which imp lies that 0UKerf which is a contradiction . Thus Kerf ≰e N and hence N is an essentially quasi-Dedekind R-module. N N M M i f g i IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Corollary (2.15) Let M be an R-module . If M is an essentially quasi-Dedekind R-module then M is an essentially quasi-Dedekind R-module . Proof : Sup p ose that M is an essentially quasi-Dedekind R-module , and since M is a quasi –injective R-module and M ≤ e M , so by (Prop 2.14 ) , M is an essentially quasi- Dedekind R-module . Corollary (2.16) Let M be an R-module . If E(M ) is an essentially quasi-Dedekind R-module then M is an essentially quasi-Dedekind R-module . Proof : It is clear . The converse of (Coro2.16) is not true in general, consider the following examp le . Exam ple (2.17) Let M = Z 2 as a Z-module . M is an essentially quasi-Dedekind Z-module. But E(Z2) = Z2 ∞ is not an essentially quasi-Dedekind Z-module , (see Rem.and.Ex 2.2(4)) . Now we p rove the following p rop osition : Proposi tion (2.18 ) Let M be an R-module such that ,for each 0,))(,(  fMEMHomf imp lies Kerf ≰e M . Then M is essentially quasi-Dedekind . Proof : Let )(MEndg R , 0g . Then ))(,( MEMHomiog  , and 0iog , where i is the inclusion map p ing . Hence Ker(iog) ≰e M . But Kerg = Ker(io g) . Thus Kerg ≰e M and M is essentially quasi-Dedekind . Next we st udy the behavior of the quotient module of essentially quasi-Dedekind module . First we have the following . Remark (2.19 ) Let M be an R-module , MN  . If M is an essentially quasi- Dedekind R-module , then NM is not necessarily essentially quasi- Dedekind R-module , consider the following examp le . Exam ple(2.20) It is well-known t hat Z as a Z-module is essentially quasi- Dedekind . Let (4)N Z  , 4(4)Z N Z Z  is not essentially quasi-Dedekind as a Z-module , ( see Rem.and. Ex 2.2(3) ) . We need to recall that an R-module P is p rojective if and only if , for any R- modules A , B and for any epimorp hism BAf : and for any homomorp hism BPg : , there exists a homomorp hism APh : such that foh = g (i.e the following diagram is a commutative) , [3 , p .117] . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Now , in the next p rop osition we give a condition under which the (Remark 2.19) is true . Proposi tion (2.21 ) Let M be an R-module such that KM is a p rojective R-module for all K ≤ e M . If M is an essentially quasi-Dedekind R-module , then NM is an essentially quasi-Dedekind R-modu le for all MN  . proof : Let NU ≤ e NM . Then U ≤ e M and hence by hyp othesis UM is a p rojective R-module . Sup p ose that there exists 0,),(  f N M NU NM Homf . But ),(),( N M U M Hom N M NU NM Hom  and since UM is p rojective , so there exists M U M g : such that πog = f , where π is the canonical p rojection map p ing . Since 0f then 0g , thus 0),( M U M Hom , U ≤ e M ; that is M is not an essentially quasi-Dedekind R-module ,which is a contradiction. Thus NM is an essentially quasi-Dedekind R-module for all MN  . A B P 0 f h g M 0 π g f U M N M IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL. 24 (3) 2011 Re ferences 1. At iy ah, M .F. and M acdonald , I.G. (1969) " Int roduction to commutative algebra " , University of Oxford . 2. Goodearl, K.R. ( 1976) " Rin g theory " M aracel Dekker , Newy ork . 3. Kasch, F. ( 1982) " M odules and rings " , Academic p ress , London . 4. Larsen, M .D. and M c Carthy , P. J. ( 1971) " M ultip lication theory of Ideals ", Academic p ress Newy ork and London . 5. M ijbass , A .S. (1997) " Quasi –Dedekind M odules " , Ph. D .T hesis , College of Science , University of Baghdad . 6. Naoum , A .G. and Hadi, I. M -A .(2002) " SQI Submodules and SQD M odules " , Iraqi J. Sci , 1 .43.D (2): 43 – 54 . 7. Rizvi , S.T. and Roman, C.S. (2007) " On K- Nonsingular M odules and app lications ", Comm . In Algebra , No.35 : 2960 – 2982 . 8. Roman, C. S . ( 2004) " B aer and Quasi - Baer M odules " , Ph . D .Thesis , Graduate , School of Ohio, State University . 2011) 3( 24بن الھیثم للعلوم الصرفة والتطبیقیة المجلدمجلة ا الواسعة معكوسة-المقاسات الجزئیة شبه الواسعةدیدیكاندیة -و المقاسات شبه ثائر یونس غاوي ، نعام محمد عليأ جامعة بغداد كلیة التربیة أبن الهیثم ،لریاضیاتقسم ا جامعة القادســیة ،كلیة التربیــة ،قسم الّریاضیات 2011حزیران 6: استلم البحث في 2011 شباط 8 :قبل البحث في ةصالخال الواســعة معكوسـة-شـبه الجزئیــة المقاسـات يمفهـوم االبحـث درســنفــي هـذا . حلقـة أبدالیــة ذا عنصـر محایـد Rلـتكن ومـن بـین . دیدیكاندیـة -شـبه و المقاسـات معكوسـة-شـبهالجزئیـة أعمام إلـى المقاسـات الواسعةدیدیكاندیة -شبه اسات والمق - مقاس غیر منفرد من النمط M اذا كان واسع دیدیكاندي -مقاس شبه M" یة تالنتائج التي حصلنا علیها النتیجة اال K " ،المقاس اذM لـنمطهو مقاس غیر منفرد مـن ا - K تشـاكل ذا كـان لكـل اf مـنM إلـىM علـى الحلقـةR Kerf ≤e Mبحیث . f = 0یؤدي إلى أن