@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 S-Coprime Submodules Inaam M.A. Hadi Rasha I. Khalaf Dept. of Mathematics/ College of Education for Pure Science (Ibn-Al-Haitham)/ University of Baghdad Received in:23 September 2012 , Accepted in: 3 February 2013 Abstract In this paper, we introduce and study the concept of S-coprime submodules, where a proper submodule N of an R-module M is called S-coprime submodule if M N is S-coprime R- module. Many properties about this concept are investigated. Key word: s-coprime submodules, coprime submodules, s-coprime modules and coprime modules. 311 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Introduction Let R be a commutative ring with unity and let M be an R-module. Annine in [1] introduce the concept coprime module, where an R-module M is called coprime if R R M ann M ann N = for all N ≨ M. Wijayanti in [2], Khalaf in [3] studied this concept. Also Ali and Khalaf in [4] introduced and studied the concept of S-coprime R-module where an R- module M is called S-coprime if R R M ann M ann N = , for all small submodules N of M (N ≪ M); that N ≪ M if N + W ≠ M for all W ≨ M. Hence it is clear that every coprime module is S-coprime module. Ali in [5], studied the concept of coprime submodule, where a proper submodule N of M is called coprime submodule if M N is coprime R-module. In this paper, we introduce the concept of S-coprime submodule, a proper submodule N of an R-module M is called S- coprime submodule if M N is S-coprime module. Moreover, we study and give the basic properties related with these concepts. Also we give many relationships between this concept and other related concepts. S.1 Basic Properties of S-Coprime Submodules First we give the following definition. Definition 1.1: Let N be a proper submodule of an R-module M. N is called S-coprime submodule if M N is S-coprime R-module. A proper ideal of a ring R is S-coprime ideal if R I is S-coprime R-module. Remarks and Examples 1.2: (1) It is clear that every coprime submodule is S-coprime submodule. However the converse is true as the following example shows: The submodule <0> in the Z-module Z is an S-coprime submodule since Z/<0> ≃ Z is S-coprime module [4, Rem. and Ex.2.1], but <0> is not coprime submodule, since Z Z 0 ≅ < > is not a coprime Z-module. (2) Every proper submodule N of a coprime R-module M is S-coprime submodule. Proof: Since M is coprime R-module, then by [5, Rem and Ex.1.1(4)], every submodule N of M is an coprime submodule. Thus the result is followed (1). (3) Any small submodule of S-coprime module is an S-coprime submodule. Proof: Let N be a small submodule of S-coprime R-module M. Then by [4, Th.12], M N is S- coprime module and hence N is S-coprime submodule Note that the converse of (3) is not true in general, for example the submodule 2< > of the Z-module Z6 (which is S-coprime) is S-coprime submodule, but it is not small. 312 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 (4) Any submodule of hollow (or chained) S-coprime R-module is a S-coprime submodule, where an R-module M is called hollow if every proper submodule of M is small [6]. An R-module M is called chained if the lattice of submodules of M is linearly ordered by inclusion [7]. Proof: It follows directly by (3). (5) If n is a square free integer then is an S-coprime submodule of the Z-module Z. Proof: If n is a square free, then by [8], Z n< > is semisimple, hence by [4, Rem. and Ex. (2)], Z n< > is S-coprime Z-module. Thus is an S-coprime submodule. (6) If W ≤ N < M such that N is an S-coprime submodule, then it is not necessarily that W is an S-coprime submodule, for example: N = <2> in the Z-module is an S-coprime submodule but W = <4> ⊂ N is not an S-coprime submodule, since 4 Z Z W  which is not an S-coprime Z-module. (7) If A is an S-coprime submodule of an R-module M and B is an S-coprime submodule in A, then it is not necessary that B is an S-coprime submodule of M as the following example shows: Consider the Z-module Z24. A = 2< > is an S- coprime submodule in Z24. B = 4< > ≤ A, B is an S-coprime submodule in A, since 2 A Z B ≅ which is an S-coprime module. However B is not an S-coprime submodule of Z24 because 24 4 Z Z B ≅ which is not an S-coprime Z-module. Recall that an R-module M is S-coprime module iff for each r ∈ R – {0}, rM ≪ M implies rM = (0), [4, Th.3]. The following results are characterizations of an S-coprime module. Proposition 1.3: Let N < M. Then the following statements are equivalent: (1) N is an S-coprime submodule. (2) For each r ∈ R – {0}, rM N M N N +  implies r ∈ [N:M]. (3) [N:M] = [W:M] for all W M N N  . Proof: (1) ⇔ (2) N is an S-coprime submodule equivalent to M N is an S-coprime R-module, which is equivalent to ∀ r ∈ R – {0}, M N M M M r r 0 N N N ⇒  by [4, Th.3], that is rM N M N N +  implies rM + N = N and hence r ∈ [N:M]. 313 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 (1) ⇔ (3) N is an S-coprime submodule means that M N is an S-coprime module, which means M N W N M ann ann N = for all W M N N  ; that [N:M] = [W:M] for all W M N N  . Corollary 1.4: Let I be an ideal of a ring R. Then I is an S-coprime ideal iff I = J for all ideal J of R s.t. J R I I  . Recall that an R-module M is called antihopfian if M M N ≅ for all N ≨ M, [9]. Proposition 1.5: Every submodule N of an antihopfian R-module is an Scoprime submodule. Proof: Since M is antihopfian module, then by [3], M is an coprime module. Hence the results follows by Rem. and Ex. 1.2(2). Remark 1.6: If A < B ≤ M such that A is an S-coprime submodule in M, then it is not necessarily that A is an S-coprime submodule in B; for example: Consider the Z-module p Z ∞ , if 2 1 Z p Α =< + > and 4 1 Z p Β =< + > . Since p Z ∞ is an antihopfian, A is an S-coprime submodule of p Z ∞ by prop.1.3. But 2pZ Β ≅ Α which is not an S-coprime module. Thus A is not an S-coprime submodule in B. The following result shows that the concepts coprime submodule and S-coprime submodule are equivalent under the class of hollow (chained) modules. Proposition 1.7: If M is a hollow (or chained) R-module, N < M. Then N is an S-coprime submodule iff N is an coprime submodule. Proof: (⇒) If M is hollow R-module, then it is clear that M N is hollow. Since N is an S-coprime submodule, then M N is an S-coprime R-module. By [4,Prop.7], M N is an coprime R-module and hence N is an coprime submodule of M. (⇐) It is clear (see Rem. and Ex. 1.2(1)). If M is a chained, then the result follows obviously, since every chained module is hollow. Recall that a proper submodule N of an R-module M is called semimaximal if M N is a semisimple R-module [10]. Remark 1.8: 314 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Every semimaximal submodule N of an R-module M is an S-coprime, but not conversely. Proof: Since N is semimaximal, then M N is a semisimple R-module, then by [4, Rem. and Ex.2(2)], M N is S-coprime module. Thus N is an S-coprime submodule. Note that <0> in the Z-module is an S-coprime submodule (see Rem. and Ex. 1.2(11)) but <0> is not a semimaximal submodule, since Z Z 0< >  is not semisimple. Proposition 1.9: Let M be an R-module, let I be an ideal of R such that I ⊆ R ann M and let N be a submodule of M. Then N is an S-coprime R-submodule of M iff N is an S-coprime R -submodule, where R R / I= . Proof: (⇒) Let N be an S-coprime R-submodule. Then M N is an S-coprime R-module. Hence by [4 , Prop.5], M N is an S-coprime R -module. Thus N is S-coprime R -module. (⇐) The proof is similarly, so it is omitted. Proposition 1.10: Let f: M → M′ be an R-epimorphism and let N < M such that N is an S-coprime submodule of M and ker f ⊆ N. Then f (N) is an S-coprime submodule of M′. Proof: Since N is an S-coprime submodule, then M N is an S-coprime R-module. To prove f (N) is an S-coprime submodule of M′, we must prove M (N)f ′ is an S-coprime R-module. Define g: M N → M (N)f ′ by g(m + N) = f (m) + N for all m + N ∈ M N . It is easy to check that g is an isomorphism; that is M M N (N)f ′ ≅ . Thus M (N)f ′ is an S-coprime R-module and hence f (N) is an S-coprime submodule of M′. Corollary 1.11: Let N, K be submodules of an R-module M such that N ⊇ K. Then N is an S-coprime submodule of M iff N K is an S-coprime submodule of M K . Proof: (⇒) Since N is an S-coprime submodule, M N is an S-coprime R-module. But M K N K M N ≅ , thus M K N K is an S-coprime R-module; that is N K is an S-coprime submodule of M K . (⇐) The proof of the converse is similarly, so it is omitted. Proposition 1.12: 315 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Let N, W be submodules of an R-module M such that W ⊇ N. If N is an S-coprime submodule of M and W ≪ M, then W is a S-coprime submodule of M. Proof: Since N is an S-coprime submodule, then M N is an S-coprime R-module. But W ≪ M implies W M N N  .Hence by [4,Th.12], M N W N is an S-coprime module. But M N W N M W ≅ , thus M N is an S-coprime module and hence W is an S-coprime submodule. Corollary 1.13: Let N ≪ M, W ≪ M. If N is an S-coprime (or W is an S-coprime) submodule of M, then N + W is an S-coprime submodule of M. Proof: By [8], N + W ≪ M. Hence the result is followed prop.1.12, directly. Next, we consider the direct sum of S-coprime submodules. Proposition 1.14: Let N1 and N2 be S-coprime submodules of R-modules M1, M2 respectively. Then N1 ⊕ N2 is an S-coprime submodule in M1 ⊕ M2. Proof: Since N1 and N2 are S-coprime submodules of M1, M2 respectively, then 1 1 M N and 2 2 M N are S-coprime R-modules. Hence by [4,Prop.18], 1 2 1 2 M M N N ⊕ is an S-coprime R-module. But 1 2 1 2 1 2 1 2 M M M M N N N N ⊕ ≅ ⊕ ⊕ , it follows that 1 2 1 2 M M N N ⊕ ⊕ is a S-coprime R-module. Therefore N1 ⊕ N2 is an S-coprime submodule of M1 ⊕ M2. S.2 S-Coprime Submodules and Multiplication Modules: First we have the following result: Proposition 2.1: Let M be a multiplication R-module and let N ≪ M. Then N is an S-coprime submodule if and only if N is a maximal small submodule of M. Proof: (⇒) Assume there exists a small submodule W such that W ⊇ N. Hence W M N N  . But N is an S-coprime submodule, [N:M] = [W:M] by prop.1.3(1⇔3). On the other hand, M is a multiplication R-module, so N = [N:M]M = [W:M]M = W. Thus N is a maximal small submodule of M. (⇐) To prove N is an S-coprime submodule. Let W M N N  . Since N ≪ M by hypothesis, so that W ≪ M. Thus W = N because N is a maximal small submodule of M. Then it is clear that [W:M] = [N:M] and hence by prop.1.3(1⇔3), N is an S-coprime submodule. Corollary 2.2: Let I ≪ R. Then I is an S-coprime ideal of R if and only if I is a maximal ideal of R. 316 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Theorem 2.3: Let M be a faithful finitely generalted multiplication R-module and let N ≨ M. Then N is an S-coprime submodule if and only if [N:M] is an S-coprime ideal. Proof: (⇒) Since N is an S-coprime submodule of M, then M N is an S-coprime R-module. But M is a multiplication module implies M N is a multiplication R-module. Hence (0) is the only small submodule in M N by [4, Rem. and Ex. 2(7)]. But M is finitely generated R- module, so M N is finitely generated. Also M N is a faithful R R R / ann M R= ≅ . It follows that R M [0 : ] R N  [11, Prop.1.1.8]; that is M ann R N  ; i.e. R [N : M] R . Again by [11, Prop.1.1.8], N ≪ M and hence by Prop.2.1, N is a maximal small submodule of M. It follows that R [N : M] is a maximal small ideal of R. To see this: suppose there exists a small ideal I of R such that I ≠ ⊃ R [N : M] . Then by [12, Th.3.1], IM ≠ ⊃ [N:M]M = N and by [12], IM ≪ M. Thus we get a contradiction, since N is a maximal small submodule of M. Therefore R [N : M] is a maximal small ideal of R and so R [N : M] is an S-coprime ideal of R. (⇐) To prove N is an S-coprime submodule, we shall prove R [N : M] = R [W : M] for all W M N N  . Since M is multiplication W = [W:M]M, N = [N:M]M, we claim that [W : M] R [N : M] [N : M]  . To see this: assume that [W : M] K R [N : M] [N : M] [N : M] + = , hence [W:M] +K = R. It follows that W + KM = M and so W KM M N N N + = .Hence KM M N N = , which implies that KM = M and hence K = R, since M is a faithful finitely generated multiplication. Thus K R [N : M] [N : M] = and so [W : M] R [N : M] I  . But [N:M] is an S-coprime ideal of R, so by Cor.1.4, [N:M] = [W:M]. Then by prop.1.3(1⇔3), N is an S-coprime submodule. Corollary 2.4: Let M be a finitely generated faithful multiplication R-module and let N < M. Then the following statements are equivalent: (1) N is an S-coprime submodule in M. (2) [N:M] is an S-coprime ideal in R. (3) N = IM for some S-coprime ideal I of R. Proof: (1) ⇔ (2) It follows by Th.2.3. (2) ⇒ (3) It is clear, since N = [N;M]M and [N:M] is an S-coprime ideal of R. (3) ⇒ (2) Since M is a finitely generated faithful multiplication module, then M = ± N and we can take I = [N:M] by [12]. Thus [N:M] is an S-coprime ideal of R. Ali in [13], introduced the concept S*-coprime module, where an R-module is an S*- coprime if for each f ∈ End(M), Im f ≪ M implies f = 0. 317 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Corollary 2.5: Let M be a finitely generated faithful multiplication R-module and let N < M. Then the following statements are equivalent: (1) N is an S-coprime submodule of M. (2) M N is an S-coprime R-module. (3) R is an S-coprime ring. (4) M is an S-coprime R-module. (5) M N is an S*-coprime R-module. Proof: (1) ⇔ (2) It is clear. (3) ⇔ (4) by [4, Prop.11]. (2) ⇔ (3) by [4, Prop.11]. (2) ⇔ (5) by [13, Prop.1.1]. Proposition 2.6: Let M be an R-module such that J(M) ≪ M. Then J(M) is an S-coprime submodule of M, where J(M) is the intersection of all maximal submodules of M, if M has maximal submodules and J(M) = M if M has no maximal submodule, [8]. Proof: It is easy to check that M J(M) has no nonzero small submodule, since J(M) ≪ M. It follows that M J(M) is an S-coprime module and hence J(M) is an S-coprime submodule. Corollary 2.7: Let M be a multiplication R-module. Then J(M) is an S-coprime submodule of M. Proof: By [12, Cor.2.6], J(M) ≪ M. Hence the result is followed Prop.2.6. Corollary 2.8: For any ring R, J(R) is an S-coprime ideal of R. S.3 S-Coprime and other Related Concepts: Proposition 3.1: If M has a finite number of maximal submodules, then J(M) is an S-coprime submodule. Proof: Let L1, L2, …, Ln be the maximal submodules of M. Then J(M) = n ii 1 L = ∩ and so that M J(M) ≅ submodule of n i 1 i M L= ⊕ [14]. It follows that M J(M) is a semisimple R-module and so S- coprime module. Thus J(M) is an S-coprime submodule. Recall that an R-module M is called local if M has a unique maximal submodule, [9]. Corollary 3.2: If M is a local ring then J(M) is an S-coprime submodule. 318 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Recall that an R-module M is called weakly supplemented if for each A ≤ M, there exists B ≤ M such that A + B = M and A ∩ B ≪ M, [15]. Proposition 3.3: If M is a weakly supplemented, then J(M) is an S-coprime submodule. Proof: Since M is a weakly supplemented R-module, then M J(M) is a semisimple R-module by [4,Rem. And Ex. 1.2(2)]. Hence M J(M) is an S-coprime module. Thus J(M) is an S- coprime submodule. Proposition 3.4: Let M be a weakly supplemented R-module and let A < M. If A is an S-coprime submodule, then there exists B ≤ M such that A + B = M, A ∩ B is an S-coprime in B. Proof: Since M is weakly supplemented and A < M, then there exists B ≤ M such that A + B = N and A ∩ B ≪ M. But A is an S-coprime submodule, so M A is an S-coprime R-module. On the other hand, M A B B A A A B + = ≅ ∩ . Thus B A B∩ is an S-coprime module and hence A ∩ B is an S-coprime submodule in B. Proposition 3.5: Let M be an S*-coprime R-module. Then every small submodule of M is an S-coprime submodule. Proof: Let N ≪ M. Since M is an S*-coprime module, then M is an S-coprime module [13,Rem. and Ex. 1.2(1)], and hence N is an S-coprime submodule by Rem. and Ex. 1.2(3). Recall that an R-module M is called scalar module if for each f ∈ End(M), there exists r ∈ R – {0}, such that f (x) = rx for all x ∈ M [16]. A ring R is called regular ring (in the sense of Von Neumann) if for each x ∈ R, there exists y ∈ R such that x = xyx [9]. To prove our next result, we prove the following lemma. Lemma 3.6: Let M be a scalar module over a regular ring R. Then M is an S-coprime module. Proof: To prove M is an S-coprime R-module. Let r ∈ R – {0} and suppose that rM ≪ M. So that to show rM = (0). Since M is a scalar R-module, then R R End(M) R / ann M≅ [17]. But R is a regular ring implies that R R / ann M is a regular ring. Thus R End(M) is a regular ring. Let f ∈ End(M) such that f (m) = rm for each m ∈ M. Then f (M) = rM. But by [9, Exc 17(a), p.272] R End(M) rM ker f= + , hence ker f = M. Since rM ≪ M. Thus f = 0 and then rM = 0. Therefore M is an S-coprime R-module. Theorem 3.7: Let M be a scalar R-module over a regular ring R. Then every small submodule of M is an S-coprime submodule. 319 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Proof: It follows by Lemma 3.6 and Rem. and Ex. 1.2(3). To prove the next result we need the following Lemma which is proved by Ali and Khalaf in [4] Lemma 3.8: Let M be a chained module over a regular ring R, then the following statements are equivalent: (1) M is an S-coprime R-module. (2) M is coprime R-module. (3) M is a prime R-module. (4) M is a quasi-Dedekind R-module. Theorem 3.9: Let M be a chained module over a regular ring and let N < M. Then the following statements are equivalent: (1) N is an S-coprime submodule in M. (2) N is a coprime submodule in M. (3) N is a prime submodule in M. We end our paper by this result: Proposition 3.10: Let N < M. If N is an S-coprime E-submodule in M, then N is an S-coprime R- submodule in M, where E = R End(M) . Proof: Since N is an S-coprime E-submodule, then M N is an S-coprime E-module. Hence by [4, Prop.2.5], M N is an S-coprime R-module. Thus N is an S-coprime R-submodule in M. References 1. Annin, S. (2002) Associated and Attached Primes Over non Commutative Rings, ph.D. Thesis, Univ. of Berkeley. 2. Wijayanti, I.E. (2006) Coprime Modules and Comodules, Ph.D. Thesis, Heinrich-Heine Universitat, Düsseldorf. 3. Khalaf, R.I. (2009) Dual Notions of Prime Submodules and Prime Modules, M.Sc. Thesis, University of Baghdad. 4. Ali, I.M. and Khalaf, R.I. (2011), S-Coprime Modules,Journal of Basrah Researchars.34. (4). 5. Ali, I.M. (2011), Coprime Submodule, Ibn Al-Haitham Journal for Pure and Applied Science,24 (2).248-256. 6. Fleury, P., (1974), Hollow Modules and Local Endomorphism Rings, Pac. J.Math.53 ,(2)..379-385. 7. Osofsky, B.L. (1991), A Contraction of Non Standard Uniserial Modules Over Valuation Domains, Bulletin Amer. Math.Soc.,25,89-97. 8. Kasch, F., (1982), Modules and Rings, Academic Press, London. 9. Hirano, Y. and Mogani, I., (1986), On Restricted Anti-Hopfian Modules, Math. J. 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Ali, I.A.M., (2006), On Ikeda-Nakayama Modules, Ph.D. Thesis, Univ. of Baghdad. 321 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 S - النمط المقاسات الجزئیة االولیة المضادة من أنعام محمد علي ھادي رشا إبراھیم خلف جامعة بغداد) /ابن الھیثم للعلوم الصرفة ( كلیة التربیة /الریاضیات علوم قسم 2013شباط 3، قبل البحث في : 2012ایلول 23: استلم البحث في الخالصة ، حیث یكون المودیول الجزئي S –في ھذا البحث قدمنا ودرسنا مفھوم المقاسات الجزئیة االولیة المضادة من النمط اذا كان S، مقاسا جزئیا اولیا ًمضادا من النمط Rعلى Mمن مقاس Nالفعلي M N . وقد S–من النمط Rمقاسا ًعلى اعطیت العدید من الخواص المتعلقة بھذا المفھوم. ، S، المقاسات الجزئیة المضادة، المقاسات المضادة من النمط Sالمقاسات الجزئیة المضادة من النمط الكلمات المفتاحیة: المقاسات المضادة. 322 | Mathematics S.1 Basic Properties of S-Coprime Submodules S.2 S-Coprime Submodules and Multiplication Modules: S.3 S-Coprime and other Related Concepts: أنعام محمد علي هادي رشا إبراهيم خلف قسم علوم الرياضيات / كلية التربية للعلوم الصرفة ( ابن الهيثم) /جامعة بغداد في هذا البحث قدمنا ودرسنا مفهوم المقاسات الجزئية الاولية المضادة من النمط – S، حيث يكون الموديول الجزئي الفعلي N من مقاس M على R، مقاسا جزئيا اوليا ًمضادا من النمط S اذا كان مقاسا ًعلى R من النمط –S. وقد اعطيت العديد من الخواص المتعلقة بهذا ...