@1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Strongly (Completely) Hollow Sub-modules II Inaam M. A. Hadi Ghaleb A. Humod Dept. of Mathematics/College of Education for Pure Science (Ibn AL-Haitham) University of Baghdad Received in:15 March 2012 , Accepted in:17 June 2012 Abstract Let M be an R-module, where R is commutative ring with unity. In this paper we study the behavior of strongly hollow and quasi hollow submodule in the class of strongly comultiplication modules. Beside this we give the relationships between strongly hollow and quasi hollow submodules with V-coprime, coprime, bi-hollow submodules. Key Words : Strongly hollow and quasi hollow submodules, strongly comultiplication modules, V-coprime module, V-coprime submodule, coprime submodule, bi-hollow module, bi-hollow submodule. 292 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 Introduction Throughout this paper, all rings are commutative rings with identity and all modules are unital module. In this research we investigate some properties of strongly hollow and quasi hollow submodules in the class of strongly comultiplication modules, see proposition 2.7, proposition 2.8. Next, we introduce the relationships between strongly hollow and quasi hollow modules with coprime modules, V-coprime modules and bi-hollow modules. Also, we study the relationships between strongly hollow and quasi hollow submodule with V-coprime (bi-hollow)-submodules. 1- Some Basic Definitions 1.1 Definition: [1,4.2] Let 0 ≠ L ≤ M, then L is called a strongly-hollow submodule (briefly, SH-submodule) if for every L1, L2 ≤ M with L ≤ L1 + L2 implies L ≤ L1 or L ≤ L2, we say that an R-module M is a strongly-hollow module if M is a strongly hollow submodule of itself. 1.2 Remark: Let 0 ≠ L ≤ M, L is a SH-submodule if for each L1, …, Ln ≤ M with L ≤ L1 + L2 + … + Ln, implies L ≤ L1 or L ≤ L2 or … or L ≤ Ln. 1.3 Definition: [1, 4.2] Let 0 ≠ L ≤ M, then L is called a completely hollow submodule (briefly, CH-submodule) if for any collection {Lλ}λ∈Λ of R-submodules of M with L Lλ λ∈Λ = ∑ , implies L = Lλ for some λ∈Λ. We say that an R-module M is completely hollow (briefly, CH-module) if M is completely hollow submodule of itself. 1.4 Definition: [2, Definition 1.13] Let 0 ≠ L ≤ M, then L is called a quasi-hollow submodule (briefly, qH-submodule) if for each L1, L2 ≤ M with L ≤ L1 + L2, then either L = L1 or L = L2. An R-module M is called a quasi-hollow module if M is a quasi-hollow submodule of itself. 1.5 Remark: [2, Remark 1.14] Let 0 ≠ L ≤ M, L is a quasi-hollow submodule if for each L1, …, Ln ≤ M with L ≤ L1 + L2 + … + Ln, then either L = L1 or L = L2 or … or L = Ln. 1.6 Definition: [3] Let M be an R-module M is called distributive if for each N, K, L ≤ M, N ∩ (K + L) = (N ∩ K) + (N ∩ L ). 1.7 Definition: [4] Let M be an R-module and let N ≤ M. N is called a strongly irreducible submodule (briefly, SI-submodule) if for each K, L ≤ M, N ⊇ K ∩ L implies either N ⊇ K or N ⊇ L. 1.8 Definition: [5] Let M be an R-module and let N ≤ M. N is called an irreducible submodule if for each K, L ≤ M, N = K ∩ L implies either N = K or N = L. 293 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 2- SH (qH,CH) Submodules and Strongly Comultiplication Modules We start this section by the following definition. 2.1 Definition: [6, Definition 2.1] An R-module M is called strongly comultiplication if I = R ann ann Μ I for every ideal I of R, and M is comultiplication, where M is comultiplication if for any L ≤ M, there exists an ideal I ≤ R such that L = M R ann ann L . Equivalently, M is strongly comultiplication if for every L ≤ M and for every ideal I ≤ R, R M I ann ann I= and L = M R ann ann L . It is clear that every strongly comultiplication is comultiplication, but the converse is not true as the following examples show: 1. Z4 as Z-module is comultiplication, see [2, Example 1.5(3)]. But it is not strongly comultiplication, since for the ideal I = <6>, I ≠ 4Z Z ann ann 6< > , because 4Z ann 6 2< >=< > and Z ann 2< > = < 2 > ≠ I. 2. Consider Z-module 3 Z ∞ is comultiplication, see [2, Example 1.5(1)]. But it is not strongly comultiplication, since if we take I = <2>, then 3 3 Z Z ann 2 0 ∞ ∞ < >= and 3 Z Z ann 0 Z ∞ = . So I ≠ 3 Z Z ann ann I ∞ . 3. Z2 as Z4-module. Z4 is comultiplication ring, but Z2 is not strongly comultiplication. Since 4 2Z Z 0 ann ann 0< >≠ < > because 2 2 Z ann 0 Z< >= and 4 2 Z ann Z 2 0=< >≠< > . 2.2 Remark: Let R be a ring. Then R is comultiplication if and only if R is strongly comultiplication. Proof: It is clear. Now we can give the following examples: 1. Zn as Zn-module is strongly comultiplication ring (comultiplication). 2. R=Z2[x,y]/ is strongly comultiplication ring. All ideals of R are x , y , x, y , xy< > < > < > < > . R R R ann x x , ann y y , ann x, y x y< >=< > < >=< > < >=< ⋅ > and R ann x y x, y< ⋅ >=< > . Recall that a ring R is QF if R self-injective ring and noetherian. Equivalently, if R is noertherian and every ideal is an annihilator (I = R R ann ann I ) [7], hence we have: 2.3 Remark: Every QF-ring is strongly comultiplication (comultiplication) ring. The following lemmas are needed in our work. 2.4 Lemma: Let M be an R-module. Let I1, I2 ≤ R. Then M ann (I1 + I2) = M ann I1 ∩ M ann I2. Proof: It is clear, so is omotted. 294 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 2.5 Lemma: Let M be a strongly comultiplication R-module, and let L1, L2 ≤ M. Then R ann (L1 ∩ L2) = R ann L1 + R ann L2. Proof: Let L1, L2 ≤ M. Since M is comultiplication, then L1 = M ann I1, L2 = M ann I2 for some I1, I2 ≤R, hence R ann (L1 ∩ L2) = R ann ( M ann (I1 + I2)) , by lemma 2.4 = I1 + I2 , since M is strongly comultiplication = R ann ( M ann I1) + R ann ( M ann I2) = R ann L1 + R ann L2 2.6 Lemma: Let M be a strongly comultiplication R-module. Let I1, I2 ≤ R. Then M ann (I1 ∩ I2) = M ann I1 + M ann I2. Proof: Since M is a stronglycomultiplication, so I1 ∩ I2 = R ann M ann (I1 ∩ I2) and I1 ∩ I2 = R ann M ann I1 ∩ R ann M ann I2 = R ann ( M ann I1 + M ann I2) Thus R ann M ann (I1 ∩ I2) = R ann ( M ann I1 + M ann I2). Hence M ann (I1 ∩ I2) = M ann I1 + M ann I2. 2.7 Proposition: Let M be a strongly comultiplication R-module. Then: (1) Every non-zero proper ideal of R is SH-ideal if and only if every non-zero proper submodule of M is SI-submodule. (2) Every non-zero proper ideal of R is qH-ideal if and only if every non-zero proper submodule of M is irreducible. Proof: (1) ⇒ Let <0> ≠ N ≨ M, N ⊇ L1 ∩ L2 where L1, L2 ≤ M. Then R ann N ⊆ R ann (L1 ∩ L2) = R ann L1 + R ann L2 , by lemma 2.5. But R ann N is a non-zero proper ideal. So by hypothesis it is SH. Hence R ann N ⊆ R ann L1 or R ann N ⊆ R ann L2. Then M ann R ann N ⊇ M ann R ann L1 or M ann R ann N ⊇ M ann R ann L2. It follows that N ⊇ L1 or N ⊇ L2. ⇐ Let <0> ≠ I ≤ R, I ⊆ I1 + I2 where I1, I2 ≤ R. Then M ann I ⊇ M ann (I1 + I2) = M ann I1 ∩ M ann I2 , by lemma 2.4. But M ann I ≠ (0) and M ann I ≨ M, so by hypothesis, M ann I is SI. Thus M ann I ⊇ M ann I1 or M ann I ⊇ M ann I2. Then R ann M ann I ⊆ R ann M ann I1 or R ann M ann I ⊆ R ann M ann I2. It follows I ⊆I1 or I⊆I2. (2) a similar proof of part (1), so is omitted. 295 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 2.8 Proposition: Let M be a strongly comultiplication R-module and let <0> ≠ N ≨ M. Then: (a) R ann N is SI-ideal if and only if N is SH-submodule. (b) R ann N is irreducible if and only if N is qH-submodule. Proof: (a) ⇒ Let N ⊆ L1 + L2 where L1, L2 ≤ M. So R ann N ⊇ R ann (L1 + L2) = R ann L1 ∩ R ann L2 Since R ann N is SI-ideal, R ann N ⊇ R ann L1 or R ann N ⊇ R ann L2. Then M ann R ann N ⊆ M ann R ann L1 or M ann R ann N ⊆ M ann R ann L2. Thus N ⊆ L1 or N ⊆ L2. ⇐ Let R ann N ⊇ I1 ∩ I2 where I1, I2 ≤ R. So M ann R ann N ⊆ M ann (I1 ∩ I2) = M ann I1 + M ann I2 Then N ⊆ M ann I1 + M ann I2. Thus N ⊆ M ann I1 or N ⊆ M ann I2, since N is a SH-submodule, so R ann N ⊇ R ann M ann I1 or R ann N ⊇ R ann M ann I2. Hence R ann N ⊇ I1 or R ann N ⊇ I2. Then R ann N is SI-ideal. (b) similar proof of part (a). Now we have several consequences of the previous proposition. 2.9 Corollary: Let M be a strongly comultiplication R-module and let <0> ≠ I ≨ R, N ≤ M. Then: (1) I is a SI-ideal if and only if M ann I is a SH-submodule. (2) I is an irreducible if and only if M ann I is qH-submodule. (3) Every non-zero submodule of M is SH-if and only if every non-zero proper ideal of R is SI. (4) Every non-zero submodule of M is qH if and only if every non-zero proper ideal of R is irreducible. Proof: (1) Since M is strongly comultiplication, I = R ann M ann I. Hence M ann I ≠ <0>. Put N = M ann I, so I = R ann N. Hence I = R ann N is a SI-ideal if and only if N = R ann N is SH, see Proposition 2.7 part (a). (2) It follow by Proposition 2.7 part (b) and a similar proof of part (1). (3) Let <0> ≠ N ≨ M. Then N = M ann R ann N. Put R ann N = I, then N = M ann I, hence I = R ann N is SI-ideal if and only if M ann I = N is SH-submodule by part (1). Thus we get the result. (4) It follows similarly from part (2). 296 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 2.10 Corollary: Let R be a comultiplication ring, <0> ≠ I ≤ R. Then: (1) I is SH-ideal if and only if R ann I is SI-ideal. (2) I is qH-ideal if and only if R ann I is irreducible. (3) I is SI-ideal if and only if R ann I is SH-ideal. Proof: It follows directly from Proposition 2.8 and Corollary 2.9. 2.11 Corollary: Let M be a strongly comultiplication and distributive R-module, let <0> ≠ N ≤ M. Then the following statements are equivalent: (1) N is SH-submodule. (2) N is qH-submodule. (3) R ann N is irreducible ideal. (4) R ann N is SI-ideal. Proof: (1) ⇔ (2) by [2, Proposition 1.16]. (1) ⇔ (3) and (1) ⇔ (4) by Proposition 2.8. 2.12 Lemma: Let M be a strongly comultiplication R-module and R is distributive. Then M is distributive. Proof: Since M ann (I1∩I2) = M ann I1 + M ann I2 by Lemma 2.6. So by [8, Lemma 3.16], M is distributive. 2.13 Corollary: Let M be a strongly comultiplication over distributive ring R, and let <0> ≠ N ≤ M. Then the following statements are equivalent: (1) N is SH-submodule. (2) N is qH-submodule. (3) R ann N is irreducible. (4) R ann N is SI-ideal. Proof: It follows from Lemma 2.12 and Corollary 2.11. 2.14 Proposition: Let M be a strongly comultiplication R-module. Then R satisfies dcc(acc) on SI-ideal if and only if M satisfies acc(dcc) on SH-submodule. Proof: ⇒ Let N1 ⊆ N2 ⊆ … be an ascending chain of SH-submodules; Ni ≠ <0> for each i = 1,2,… . So R ann Ni is SI-ideal for each i = 1,2,… , see Proposition 2.5 part (a). But R ann N1 ⊇ R ann N2 ⊇ … . So by dcc on SI-ideal of R, there exists n ∈ Z+ such that R ann Nn = R ann Nn + 1 = … . Then M ann R ann Nn = M ann R ann Nn + 1 = … . Thus we get Nn = Nn + 1 = … . ⇐ The proof is similarly. 297 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 By a similar proof R satisfies acc on SI-ideals if and only if M satisfies dcc on SH- submodules. 3- SH(qH) and V-Coprime (Bihollow) Submodules Recall that an R-module M is called coprime module if R ann M = R M ann N , for every proper submodule N of M, see [9]. Equivalently, M is coprime if for every non-zero ideal I of R, either IM = <0> or IM = M, see [10]. A proper submodule K of M is called coprime submodule in M if M/K is a coprime R- module, see [1, Proposition 3.10]. 3.1 Remark: The concept of coprime module and SH(qH)-module are independent as the following examples show: (1) The Z-module Q is coprime module, since for every r ∈ Z, either rQ = <0> or rQ = Q. But Q is not SH(not qH), see [2, Remark 1.4(3)]. (2) Z4 as Z-module is SH(qH). But Z4 is not coprime Z-module, since 2Z4 = 2< > ≠ Z4 and 2Z4 ≠ 0< > . Recall that a proper submodule N of an R-module M is called invariant if for each f ∈ EndRM, f (N) ⊆ N,see [11]. Invariant submodule is called fully invariant submodule by some authors such as see [10]. In 2005, [12] define coprime submodules and modules as follows: 3.2 Definition: Let K ≤ M be a fully invariant submodule, K is called a coprime submodule of M if for any fully invariant submodules L, L' ≤ M with K ⊆ (L M : L') implies K ⊆ L or K ⊆ L' where (L M : L') = ∩{f – 1(L'); f ∈ EndRM, f (L) = <0>}. M is called coprime module if M coprime submodule of itself. 3.3 Lemma: Let L, L' be fully invariant submodules of an R-module M. Then L + L' ⊆ (L M : L'), see [12, 4.1(ii)]. 3.4 Remark: The concept of coprime submodules (in the sense of J.Abu.), and (in the sence of J.Rios) are independent as the following examples show: (1) Consider the Z-module Z, let N = <3>. N is coprime submodule in Z (in the sense of J.Abu) since Z/N ≃ Z3 which is a coprime Z-module. But N is not coprime submodule (in the sense of J.Rios), because N = <3> ⊆ <4> + <5> = Z. But by previous lemma <4> + <5> ⊆ (<4> Z : <5>). So N ⊆ (<4> Z : <5>) and N ⊈ <4>, N ⊈ <5>. 298 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 (2) Consider the Z-module Z8. Let N = 4< > . N is not coprime submodule (in the sense of J.Abu.), because Z8/ 4< >≃ Z4 which is not coprime Z-module. Now N ⊆ 2 0< > + < > , and by lemma 3.3, 8Z 2 0 ( 2 : 0 )< > + < >⊆ < > < > , so N ⊆ 8Z ( 2 : 0 )< > < > and N ⊆ 2< > . Similarly, N 8Z 4 0 ( 4 : 0 )⊆< > + < >⊆ < > < > and N ⊆ 4< > . Thus, for any L, L' ≤ Z8, N⊆ (L M : L'), implies N ⊆ L or N ⊆ L'. Then N is coprime submodule (in the sense of J.Rios). 3.5 Remark: The concept of coprime module (in the sense of S.Annin,2002) and (in the sense of J.Rios, 2005) are independent as the following example shows: The Z-module Z4 is not coprime module (in the sense of S.Annin), see Remark 3.1(2). But Z4 is coprime Z-moduule (in sense of J.Rios) as follows: Suppose Z4 ⊆ (L 4Z : L'). Since (L 4Z : L') is a submodule of Z4, then (L 4Z : L') = Z4. But by simple calculation, we have: 4 4 4 4( 0 : 0 ) 0 , ( 0 : 2 ) 2 , ( 0 : Z ) Z , (Z : 2 ) Z< > < > =< > < > < > =< > < > = < > = , ( 2 : 0 ) 2< > < > =< > . Thus if (L 4Z : L') = Z4, then either L = Z4 or L' = Z4; that is Z4 is coprime Z-module (in the sense of J.Rios). 3.6 Remark: The concept of coprime module (in sense of J.Rios) and SH(qH)-module are independent as follows: The Z-module Q is coprime (in sense of J.Rios) by [12, Remark 4.7(2)]. But Q is not SH(qH) as Z-module by [2, Remark 1.4(3)]. In 2005, [12] introduced the concept of V-coprime and bi-hollow submodules as follows : 3.7 Definition: If K, L, L' are fully invariant submodules of M. Then: (1) K is called V-coprime if K ⊆ L + L' implies K ⊆ L or K ⊆ L'. (2) K is called bi-hollow if K = L + L' implies K = L or K = L'. (3) M is called bi-hollow if M = L + L' implies M = L or M = L'. 3.8 Remark: If K is V-coprime submodule of an R-module M, then K is bi-hollow. Proof: Let K = L + L' where L, L' are fully invariant submodules of M. So K ⊆ L + L' and since K is V-coprime, then K ⊆ L or K ⊆ L'. But K ⊇ L + L'. Hence K = L or K = L'. 3.9 Proposition: M is bi-hollow R-module if and only if M is V-coprime. Proof: ⇒ Let M ≤ L + L' where L, L' are fully invariant submodules of M. So M = L + L', then M = L or M = L' since M is bi-hollow. Hence M ⊆ L or M ⊆ L'. ⇐ Let M = L + L', then M ⊆ L + L'. So M ⊆ L or M ⊆ L'. Since M is V-coprime. Hence M = L or M = L'. 299 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 3.10 Proposition: If K is coprime submodule of an R-module M (in the sense of J.Rios), then K is V- coprime. Proof: Let K ⊆ L + L' where L, L' are fully invariant submodules of M. But L + L' ⊆ (L M : L') by Lemma 3.3. So K ⊆ (L M : L') and since K is coprime submodule of M, hence either K ⊆ L or K ⊆ L'. Thus K is V-coprime, hence K is bi-hollow. 3.11 Remark: The concept of V-coprime (bi-hollow) does not implies SH(qH) modules as the following examples shows: The Z-module Q is coprime (in sense J.Rios), then Q is V-coprime (bi-hollow) by previous Proposition. But Q is not SH (not qH) by [2, Remark 1.4(3)]. 3.12 Lemma: Let M be an R-module. If M is a multiplication (or cyclic or scalar or comultiplication). Then every submodule of M is fully invariant. Proof: It is known if M is a multiplication, then every submodule is fully invariant of M. If M is cyclic, then M is multiplication, so every submodule is fully invariant. If M is scalar R-module, then for each f ∈ EndRM, there exists r ∈ R such that f (x) = rx for all x ∈ M. Hence, if N ≤ M, then f (N) = rN ⊆ N; that is N is fully invariant. If M is comultiplication, then every submodule is fully invariant, see [8, Theorem 3.17(a)]. 3.13 Corollary: Let M be a multiplication (or cyclic or scalar or comultiplication), let N be a non-zero submodule of M. Then (1) N is V-coprime if and only if N is SH-submodule. (2) N is bi-hollow if and only if N is qH-submodule. (3) M is bi-hollow if and only if M is SH-submodule. Proof: It follows directly from the previous lemma and definitions of SH(qH) and V-coprime (bi-hollow) submodules. 3.14 Remark: The concept of coprime submodule (in sense of J.Abu.) and V-coprime submodule are independent as the following examples show: (1) Consider the Z-module Z. Let N = <3> ⊆ Z. N is coprime submodule of Z (in sense of J.Abu) by Remark 3.4(1). But N is not SH. Hence N is not V-coprime by Corollary 3.13. Since Z is multiplication Z-module. (2) Consider the Z-module Z8. Let K = 4< > is a V-coprime submodule of Z8 as Z-module. But K is not coprime submodule (in sense of J.Abu), because Z8/K ≃ Z4 which is not coprime module (in sense of S,Annin or Wij.). Now we investigate the behaviour of SH(qH)-submodule under the Localization. 300 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 3.15 Proposition: Let N be a submodule of an R-module M, and S is multiplicatively closed subset of R. Then S – 1N is SH(qH-submodule) of S – 1M as S – 1R-module if and only if N is SH(qH- submodule) of M. [Provided, S – 1N ⊆ S – 1W ⇔ N ⊆ W]. Proof: ⇒ If S – 1N is qH, let N = L1 + L2 where L1, L2 ≤ M. So S – 1N = S – 1(L1 + L2). Then S – 1N = S – 1L1 + S – 1L2, so S – 1N = S – 1L1 or S – 1N = S – 1L2 since S – 1N is qH. Hence N = L1 or N = L2. ⇐ If N is qH, let S – 1N = S – 1L1+ S – 1L2 , where S – 1L1, S – 1L2 ≤ S – 1M = S – 1(L1 + L2). Then S – 1N = S – 1(L1 + L2) implies N = L1 + L2. So N = L1 or N = L2 by hypothesis. S – 1N = S – 1L1 or S – 1N = S – 1L2. By a similar proof, N is SH iff S – 1N is SH. References 1. Abuhlail J.Y. (2011) Zariski Topologies for Coprime and Second Submodules, Deanshib of Scientific Research at King Fahad University of Peroleum and Minerals, February 4. 2. Hadi, I.M.A. and Humod, GH.A (2012) Strongly (Completely) hollow Submodules I, Ibn- Al-Haitham Journal for Pure and Applied Science, to appear. 3. Barnard, A. (1981) Multiplication Modules, J.Algebra, 71:174-178. 4. Bourbaki, N. (1998) Commutative Algebra, Springer-Verlag. 5. Dauns, J. (1980) Prime Modules and One-Sided Ideals in Ring Theory and Algebra III, Proceeding of the Third Oklahoma Conference, B.R.MCDonald (editor), Dekker, NewYork, p.301-344. 6. Ansari Toroghy, H. and Farshadifar, F. (2009) Strongly Comultiplication Modules, CMU J.Nat.Sci., 8(1):105-113. 7. Ikeda, M. and Nakayama,T. (1954) On Some Characterstic Properties of Quasi-Frobenius and Regular Ring, Proc. Amer. Math. Soc, 5:15-19. 8. Ansari Toroghy, H. and Farshadifar, F. (2007) The Dual Notion of Multiplication Modules, Taiwanese J.Math., 11(4):1189-1201. 9. Annin, S. (2002) Associated and Attached Primes Over Non Commutative Rings, Ph.D. Dissertation University of California at Berkeley. 10. Wijayanti, I. (2006) Coprime Modules and Comodules, Ph.D. Dissertation, Heinrich-Hein University, Dusseldorf. 11. Faith, C. (1973) Rings: Modules and Categories I, Springer-Verlage, Berline, Heidelberg, New York. 12. Raggi, F. Rios Montes, J. and Wisbauer, R. (2005) Coprime Preradicals and Modules, J.Pur.App.Alg., 200:51-69. 301 | Mathematics @1a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹26@@ÖÜ»€a@I1@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (1) 2013 IIالمقاسات الجزئیة المجوفة (التامة) بقوة ھادي انعام محمد علي غالب أحمد حمود جامعة بغداد / )ابن الھیثم( للعلوم الصرفةكلیة التربیة /قسم الریاضیات 2012حزیران 17قبل البحث في: ، 2012اذار 15استلم البحث في: الخالصة حلقة ابدالیة ذات محاید. في ھذا البحث درسنا سلوك كل من المقاسات الجزئیة Rحیث Rمقاسا ً على Mلیكن المجوفة بقوة وشبھ المجوفة في المقاسات الجدائیة المضادة بقوة. اضافة الى ھذا درسنا العالقات بین المقاسات الجزئیة والمقاسات الجزئیة المضادة والمقاسات Vالمجوفة بقوة وشبھ المجوفة مع المقاسات الجزئیة االولیة المضادة في النمط المجوفة الثنائیة. المقاسات الجزئیة المجوفة بقوة وشبھ المجوفة ، المقاسات الجدائیة المضادة بقوة، المقاسات االولیة : الكلمات المفتاحیة المضادة، المقاسات المجوفة ، المقاسات الجزئیة V، المقاسات الجزئیة االولیة المضادة من النمط Vالمضادة من النمط الثنائیة، المقاسات الجزئیة المجوفة الثنائیة. 302 | Mathematics