Microsoft Word - 164-178      164 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31المجلد مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 W-Closed Submodule and Related Concepts Haibat K. Mohammad Ali Mohammad E. Dahsh Dept. of Mathematics/College of Computer Science and Mathematics/Tikrit University Dr .mohammadali2013@gmail.com Mohmad.alduri90@gmail.com Received in:30/January/2018, Accepted in:7/March/2018 Abstract Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied. Keywords: Closed submodules, Weak essential submodules, W-closed submodules, completely essential modules, y-closed submodules, Minimal semi-prime submodules.      165 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Introduction In this note, we shall assume that all rings are commutative with unity and all modules are unital left modules, and all R-modules under study contains semi-prime submodules. "A submodule L of a module M is called closed in M provided that L has no proper essential extension in M [1]" ," where a non-zero submodule N of M is called essential if N ∩ E 0 for all non-zero submodule E of M [1]", "and a non-zero submodule N of M is called weak essential if N ∩ S 0 ∀ non zero semi-prime submodule S of M [2]". "Equivalently, a submodule N of a module M is called weak essential if whenever N ∩ S 0 , then S=(o) for every semi-prime submodule S of M [3]","where a submodule S of a module M is called semi-prime if for each 𝑟 ∈ 𝑅 𝑎𝑛𝑑 𝑦 ∈ 𝑀 𝑤𝑖𝑡ℎ 𝑟 𝑦 ∈ 𝑆, 𝑘 ∈ 𝑍 𝑡ℎ𝑒𝑛 𝑟𝑦 ∈ 𝑆 [4]"."Equivalently if 𝑟 𝑦 ∈ 𝑆, 𝑡ℎ𝑒𝑛 𝑟𝑦 ∈ 𝑆 [5]".In this proper, "we introduce the concept of w-closed submodule "which is stronger than the concept of closed submodule" ,where a submodule K of an R-module M is called w-closed "if K has no proper weak essential extension in M". That is if K is weak essential in L , where L is a submodule of M, then K=L . A module M is called chaine if for each submodules E and D of M either 𝐸 ⊆ 𝐷 𝑜𝑟 𝐷 ⊆ 𝐸 [6]. An R-module M is called fully semi-prime, if every proper submodule of M is semi-prime submodule [3].A semi-prime radical of a module M denoted by Srad( M ), and it is the intersection of all semi-prime submodule of M [3]. A submodule N of a module M is called y-closed submodule in M, if is a non-singular module [1],"where an R- module M is called non-singular if 𝑍 𝑀 𝑥 ∈ 𝑀: 𝑎𝑛𝑛 𝑥 𝑖𝑠 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 𝑖𝑑𝑒𝑎𝑙 𝑖𝑛 𝑅 =(0) [3]". A module M is called multiplication module, if every submodule N of M is equal IM. i.e N=IM for some ideal I of R [7]. Basic Properties of W-Closed Submodules "In this section, we introduce the definition of" w-closed submodule, and we will give basic properties, examples of w-closed submodule. Definition (2.1) Asubmodule K of a module M is called w-closed in M ,"if K has no proper weak essential extension in M". That is if there exists asubmodule L of M with K "is a weak essential submodule of L", then K=L . An ideal J of R is called w-closed, if it is w-closed R- submodule. Remark (2.2) Every w-closed submodule in a module M is a closed submodule in M,but the converse is not true in general. proof Let K be a w-closed submodule in M and L is a submodule in M with K is essential in L, then by [2] K is weak essential in L. But K is w-closed in M, thus K=L. Hence K is closed submodule in M. For the converse, we give the following example: 3)(2.Example Let M=𝑍 as a Z-module, and 𝐾 〈3〉 is closed submodule in 𝑍 , since K is a direct summand of the Z-module 𝑍 , but K is not w-closed submodule in 𝑍 because K is weak essential submodule in 𝑍 . Proposition (2.4) If M is a module, and E is a submodule of M such that E is weak essential and w-closed in M,then E=M.      166 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proof Follows from definition of w-closed submodule. Remark (2.5) (1) Every module M is a w-closed submodule in itself. (2) The trivial submodule <0> may not be w-closed submodule of an R-module M, for example : 𝑀 𝑍 as a Z-module, 𝐾 〈0〉 is not w-closed submodule in M. Proposition(2.6) If M is a module, and let U be a non-zero submodule of M, then ∃ a w-closed submodule T in M with U is weak essential in T. proof Let 𝒜 ={ Q : Q "is a submodule of M such that" U is weak essential in Q }. clearly 𝒜 is a non-empty. 𝒜 has maximal element say T "by Zorn's lemma”. "To prove that" T is a w- closed submodule in M. Assume that there exists a submodule L of M with T weak essential in L. Since U is weak essential in T and T is weak essential in L so by [3, prop (1.4)]. U is weak essential in L. But this is a contradicts the maximality of T.Thus T=L. Hence T is w- closed submodule in M, with U is weak essential in T. The following remark shows that w-closed property is not hereditary property. Remark(2.7) If 𝑄 and 𝑄 are submodules of an R-module M with 𝑄 is a submodule of 𝑄 , and 𝑄 is a w-closed submodule in M then 𝑄 need not to be w-closed submodule in M. For example: M=Z the Z-module, M is a w-closed submodule of M, and 2Z is a submodule of M is not w- closed submodule in M, since 2Z has a proper weak essential extension. The converse of remark (2.7) is not true. That is if 𝑄 is w-closed in M, then 𝑄 need not to be w-closed in M. As the next example explain: Example(2.8) Take the Z-module Z and 𝑁 = <0>, 𝑁 = 2Z are Z-submodules of Z we notes that 𝑁 is w-closed submodule in Z. But 𝑁 is not w-closed submodule in Z. The following propositions show that the transitive property for w-closed submodule hold under certain conditions. Proposition (2.9) If E and D are submodules of a module M, provided that D contained in any weak essential extensions of E, and E is a w-closed submodule in D and D is a w-closed submodule in M, then E is a w-closwed submodule in M. Proof Assume that K is a submodule of M such that E is weak essential in K. By hypothesis D is a submodule of K. Since E "is weak essential in K and E is a submodule of D" then by [2, Rem(1.5)(2)] we get D is weak essential in K. But D is w-closed submodule in M, then D=K. That is E weak essential in D. But E is w-closed submodule in D, so E=D. Hence E is a w- closed submodule in M.      167 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition(2.10) If 𝑁 and 𝑁 are submodules of a module M, provided that 𝑁 is containing any weak essential extensions of 𝑁 , and 𝑁 is a w-closed submodule in 𝑁 and 𝑁 is a w-closed submodule in M, then 𝑁 is a w-closed submodule in M. Proof Assume that 𝑈 𝑀 with 𝑁 is weak essential submodule in U, then by hybothesis we get U is a submodule in 𝑁 . Since 𝑁 is a w-closed in 𝑁 , then 𝑁 =U. Thus 𝑁 is a w-closed .submodule in M Proposition(2.11) If M is a chained module, and E, D are submodules of M with 𝐸 𝐷, and 𝐸 𝐷 and 𝐷 𝑀, then 𝐸 𝑀. Proof Let K be a submodule of M with E is weak essential in K. Since M is chained module, then either K is a submodule in D or D is a submodule in K. If K is a submodule in D, and since E is a w-closed submodule in D,then E=K.Hence E is a w-closed submodule in M. If D is a submodule in K, and since E is weak essential in K, then by [2, Rem(1.5)(2)] D is a weak essential submodule in K. But D is a w-closed submodule in M, hence D=K.Thus, E is a weak essential submodule in D. But E is a w-closed submodule in D, then E=D.Hence E is a w- closed submodule in M. Before we give the next proposition, we introduce the following denifition. "Definition(2.12) A module M is called completly essential if every non zero weak essential submodule of M is an essential submodule of M". Completely essential in [3] is called fully essential. The following proposition show that closed submodules and w-closed submodules are equivalents under certain conditions. Proposition(2.13) "If M is a module, and E be a non zero submodule of M" such that every weak essential extensions of E is a completly essential, then E is a closed submodule in M if and only if E is a w-closed submodule in M. Proof Let E be a non zero closed submodule in M, and U be a submodule of M such that E is a weak essential in U. By hypothesis U is a completely essential, therefore E is an essential submodule in U. But E is a closed submodule in M, then E=U.That is E is a w-closed submodule. The converse is direct. Proposition(2.14) If M is a fully semi-prime module, and E be a non zero submodule of M, then E is a closed submodule in M if and only if E is a w-closed submodule in M.      168 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proof Assume that E is a non zero closed submodule in M, and U is a submodule of M such that E is a weak essential submodule in U. Then by [3, Cor(2.5)] E is an essential submodule in U. But E is a non-zero closed submodule in M, hence E=U. That is E is a w-closed submodule in M. The converse is direct. Corllary (2.15) If M is a uniform module, and E be a non zero submodule of M, then E is a closed submodule in M if and only if E is a w-closed submodule in M. Proof Assume that E is a closed submodule in M and let E a weak essential in U where U is a submodule of M, then U is a uniform. Hence by [3,prop(2.7)] U is a completely essential. Thus E is an essential in U. But E is a closed, then E=U. Thus E is a w-closed in M. The converse is direct. The following propositions show that the transitive property for w-closed submodules hold under conditions fully semi-prime and completely essential. Proposition(2.16) Let M be a module, and E, D are non-zero submodules of M such that 𝐸 𝐷 and every weak essential extensions of E is a completely essential submodule of M. If 𝐸 𝐷 and 𝐷 𝑀, then 𝐸 𝑀. Proof Since 𝐸 𝐷 and 𝐷 𝑀. Then by remark(2.2), we get E is a closed submodule in D and D is a closed submodule in M. Then by [1,prop(1.5),P.18] "we get E is a closed submodule in M", then by prop(2.13), 𝐸 𝑀. Proposition (2.17) Let M be a fully semi-prime module, and let E be a non-zero w-closed submodule in D and D is a w-closed submodule in M. Then E is a w-closed submodule in M. Proof Since E is a w-closed submodule in D and D is a w-closed submodule in M, then by remark(2.2), E is a closed submodule in D and D is a closed submodule in M. Hence by [1,prop(1.5), P.18] we get E is a closed submodule in M. Thus by prop(2.14), E is a w-closed submodule in M. Remark (2.18) The intersection of two w-closed submodule need not to be w-closed submodule as the following example shows: In the Z-module 𝑍 ⨁ 𝑍 , the submodules 𝑁 〈 0, 1 〉 𝑎𝑛𝑑 𝐾 〈 4, 1 〉 are w-closed submodule in 𝑍 ⨁ 𝑍 , but 𝑁 ∩ 𝐾 0, 0 is not w-closed submodule in 𝑍 ⨁ 𝑍 . The following results give more basic properties of w-closed submodules.      169 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition (2.19) If every submodule of a module M is w-closed, then every submodule of M is a direct summand. Provided that M is a semi simple. Proof Since every submodule of M is w-closed, then every submodule of M is a closed. Hence by [8, Exc(6-c), P.139] "every submodule of M is a direct summand of M". The following corollary is a direct consequence of proposition(2.19). Corollary (2.20) If every submodule of a module M is a w-closed, then M is a semi-simple. Proposition(2.21) If E and D are submodules of a module M with 𝐸 𝐷, and 𝐸 𝑀, then 𝐸 𝐷. Proof Let 𝐹 𝐷, then 𝐹 𝑀, and E is a weak essential submodule of F. But 𝐸 𝑀, then E=F. Hence 𝐸 𝐷. As a direct application of proposition(2.21) we get the following results. Corollary (2.22) If E and D are submodules of a module M with 𝐸 ∩ 𝐷 is a w-closed submodule in M , then 𝐸 ∩ 𝐷 is a w-closed submodule in E and D. Corollary (2.23) If M is a module, and E , U are w-closed submodules in M,then E and U are w-closed submodules in E + U. Corollary(2.24) If M is an R-module , and E is a w-closed submodule in M, then E is a w-closed submodule in √𝐸. Proof Since 𝐸 √𝐸 𝑀, and E is a w-closed submodule in M then by proposition(2.21), E is a w-closed submodule in √𝐸. Remark (2.25) A direct summand of a module M is not necessary w-closed submodule in M, as the following example show: Let M=𝑍 as a Z-module , where 𝑍 〈3〉 ⨁ 〈8〉, the direct summand 〈3〉 is not w-closed submodule in 𝑍 . Since 〈3〉 is a weak essential in 𝑍 . Proposition(2.26) Let 𝑋 𝑋 ⨁ 𝑋 be a module, where 𝑋 and 𝑋 are submodules of X, and let E be a non zero w-closed submodule in 𝑋 and D is a non zero w-closed submodule in 𝑋 such that ann 𝑋 + ann 𝑋 =R, and all weak-essential extensions of 𝐸 ⨁ 𝐷 are completely essential submodule of 𝑋 ⨁ 𝑋 . Then 𝐸 ⨁ 𝐷 is a w-closed submodule in 𝑋 ⨁ 𝑋 .      170 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proof Let 𝑆 𝑋 with 𝐸 ⨁ 𝐷 "is a weak essential submodule in S". Since S is a submodule of X and ann 𝑋 + ann 𝑋 =R, then by [9, prop(4.2)], 𝑆 𝑆 ⨁ 𝑆 , where 𝑆 is a submodule of 𝑋 and 𝑆 is a submodule of 𝑋 . Thus 𝐸 ⨁ 𝐷 is a weak essential submodule in 𝑆 ⨁ 𝑆 . But by hybothesis S is a completely essential , therefore 𝐸 ⨁ 𝐷 is an essential submodule in 𝑆 𝑆 ⨁ 𝑆 , thus by [10, prop(5.20)] we are, "E is an essential submodule in 𝑆 and D is an essential submodule in 𝑆 ". Since both E and D are w-closed, it is a clear that E and D are closed submodules in 𝑆 and 𝑆 respectively. Then E=𝑆 and D=𝑆 , thus 𝐸 ⨁ 𝐷=𝑆 ⨁ 𝑆 . That is 𝐸 ⨁ 𝐷 is a w-closed submodule in X. Proposition(2.27) Let 𝑋 𝑋 ⨁ 𝑋 be a module, where 𝑋 and 𝑋 are submodules of X such that ann 𝑋 + ann 𝑋 =R and all submodules of X are completely essential submodule of X. If E and D are non zero submodules of 𝑋 and 𝑋 respectively, then 𝐸 ⨁ 𝐷 is a w-closed submodule in X if and only if E is a w-closed submodule in 𝑋 and D is a w-closed submodule in 𝑋 . Proof ⟸ Suppose that 𝐸 ⨁ 𝐷 "is weak essential submodule of K", "where K is a submodule of M". Hence by [1, prop(4.2)] 𝐾 𝐾 ⨁ 𝐾 where 𝐾 is a submodule of 𝑋 and𝐾 is a submodule of 𝑋 . Thus 𝐸 ⨁ 𝐷 is weak essential submodule in 𝐾 ⨁ 𝐾 . But 𝐾 ⨁ 𝐾 is a completely "essential submodule of " X, then 𝐸 ⨁ 𝐷 "is an essential submodule of " 𝐾 ⨁ 𝐾 . Hence by [10, prop(5.20), P.15] we get "E is an essential submodule in 𝐾 and D is an essential submodule in 𝐾 ". But by [2] every essential submodule is a weak essential. Hence E "is a weak essential submodule in 𝐾 " and D is a weak essential submodule in 𝐾 . But E and D are w-closed submodules of X, then E=𝐾 and D=𝐾 . Thus 𝐸 ⨁ 𝐷=𝐾 ⨁ 𝐾 . That is 𝐸 ⨁ 𝐷 is a w-closed submodule in X. ⟹ Assume that E "is a weak essential submodule in L" where L is a submodule of X, we have D is a weak essential submodule in D. But by hypothesis all submodules of X are completely essential, then E is an essential submodule in L and D is an essential submodule in D. Hence by [10, prop(5.20), P.15], we have. 𝐸 ⨁ 𝐷 is an essential submodule in 𝐿 ⨁ 𝐷, which implies that 𝐸 ⨁ 𝐷 is a weak essential submodule in 𝐿 ⨁ 𝐷. Hence 𝐸 ⨁ 𝐷= 𝐿 ⨁ 𝐷. That is E=L, implies that E is a w-closed submodule in𝑋 . In similar way we can prove that D is w-closed submodule in𝑋 . It is well-known that a fully semi-prime module is a completely essential [3, cor(2.6)]. So we have the following result. Corollary(2.28) If 𝑋 𝑋 ⨁ 𝑋 is a module, where 𝑋 and 𝑋 are submodules of X with ann 𝑋 + ann 𝑋 =R and all submodules of X are fully-semi-prime. If E, D are submodules of 𝑋 and 𝑋 respectively, then 𝐸 ⨁ 𝐷 is a w-closed submodule in X if and only if E is a w-closed submodule in 𝑋 and D is a w-closed submodule in 𝑋 . The following remark shows that w-closed property is not algebrice property. Remark(2.29) If M is a module, and X is a w-closed submodule of M, and Y is asubmodule of M such that X≅Y , then it is not necessary that Y is a w-closed submodule in M, as the following example      171 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 shows:- The Z-module Z is a w-closed in itself and Z ≅ 3Z, but 3Z as a Z-module is not a w- closed submodule in Z, since 3Z "is a weak-essential submodule of Z". We introduce the following lemma, before we give the next proposition. Lemma(2.30) Let 𝑓 ∈ 𝐻𝑜𝑚 𝑀 , 𝑀 be module an epimorphism with 𝐾𝑒𝑟 𝑓 𝑆𝑟𝑎𝑑 𝑀 , if 𝐸 𝑀 . Then 𝑓 𝐸 𝑤𝑒𝑎𝑘 𝑀1 . Proof Assume that 𝐸 𝑀 , and 𝑓 𝐸 ∩ 𝑆 0 where S is a semi-prime submodule of 𝑀 . But 𝐾𝑒𝑟 𝑓 𝑆𝑟𝑎𝑑 𝑀 𝑆 for all semi-prime submodule S of 𝑀 , hence by [5, prop(2.1)(A)] 𝑓 𝑆 is a semi-prime submodule of 𝑀 . That is 𝐸 ∩ 𝑓 𝑆 0 , but E "is a weak essential submodule of 𝑀 ", then 𝑓 𝑆 0 . Implies that 𝑆 𝐾𝑒𝑟𝑓 𝑓 𝐸 , and hence 𝑓 𝐸 ∩ 𝑆 0 implies that 𝑆 0 . Then 𝑓 𝐸 𝑤𝑒𝑎𝑘 𝑀1. Proposition(2.31) Let 𝑔: 𝑀 → 𝑀 be a module epimorphism, and let E be a submodule of 𝑀 such that ker 𝑔 𝑆𝑟𝑎𝑑 𝑀 ∩ 𝐸.If E is a w-closed submodule in𝑀 then 𝑔 𝐸 is a w-closed submodule in 𝑀 . Proof Suppose that E is a w-closed submodule in 𝑀 , and let 𝑔 𝐸 "is a weak essential submodule of L", where L is a submodule of 𝑀 . Since ker 𝑔 𝑆𝑟𝑎𝑑 𝑀 ∩ 𝐸. Hence by lemma(2.30), we get 𝑔 𝑔 𝐸 is a weak essential submodule in 𝑔 𝐿 , where 𝑔 𝐿 is a submodule of 𝑀 , but 𝐾𝑒𝑟𝑔 𝐸, then 𝑔 𝑔 𝐸 𝐸, i.e E is a weak essential in 𝑔 𝐿 . But E is a w- closed submodule in 𝑀 , then E=𝑔 𝐿 , and since 𝑔 𝑖𝑠 an epimorphism so , 𝑔 𝐸 𝐿. Hence 𝑔 𝐸 is a w-closed submodule in 𝑀 . As a direct consequence of proposition(2.31) we get the following corollary. Corollary(2.32) : If E and D are submodules of a module M with 𝐸 𝑠𝑟𝑎𝑑 𝑀 ∩ 𝐷. If D is a w-closed submodule in M, then is a w-closed submodule in . The following proposition gives a relation between y-closed submodule and w-closed submodule in the class of a fully semi-prime module. Proposition (2.33) Let M be a fully semi-prime module. Then every non zero y-closed submodule is a w- closed submodule. Proof Let E be a non zero y-closed submodule in M, then by [11], every y-closed submodule is a closed. Hence E is a closed, then by proposition(2.14), E is a w-closed submodule in M. "The following proposition shows that in the class of non-singular modules", the class of w- closed submodules is contained in the class of y-closed submodules. Proposition (2.34) If M is a non singular module and E is a w-closed submodule of M, then E is a y-closed submodule of M.      172 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proof Let E be a w-closed submodule in M then E "is a closed submodule in M", but M is a non-singular R-module, then by [11, prop(2.1)(2)] E is a y-closed submodule in M. The following proposition shows that in the class of non-singular and fully semi-prime R- module, w-closed submodule , y-closed submodule and closed submodule are equivalent: Proposition (2.35) Let M be a fully semi-prime and non-singular module, "and E be a non zero submodule of M. Then the following statements are equivalent" : 1- E is a y-closed submodule . 2- E is a closed submodule . 3- E is a w-closed submodule. Proof 𝟏 ⟹ 𝟐 Follows by [11]. 𝟐 ⟹ 𝟑 Follows by proposition(2.14). 𝟑 ⟹ 𝟏 Follows by proposition(2.34). 3. W-closed submodule in multiplication modules In this section, we establishe some relationships between w-closed submodule and multiplication modules. "First we introduce the following definition". Definition(3.1) A non-zero semi-prime submodule E of a module M is called minimal semi-prime submodule of M, if whenever S "is a non zero semi-prime submodule of M such that" 𝑆 𝐸, then S=E. That is by minimal semi-prime submodule E of M we mean a semi-prime submodule which is a minimal in the collection of semi-prime submodules of M. If A is a proper ideal of R, then a semi-prime ideal B is called a minimal semi-prime ideal of A provided that 𝐴 𝐵 and is minimal semi-prime ideal of a ring . Remark(3.2) In multiplication module since 𝑎𝑛𝑛 𝑀 𝑅 it follows that by [12, Th(2.5)], there exists a minimal ideal P of R such that 𝑎𝑛𝑛 𝑀 𝑃, 𝑎𝑛𝑑 𝑀 𝑃𝑀. But by [13, prop(2.5), P.36] PM is a semi-prime submodule of M. Then from definition(3.1) we get the following facts: (a) E is a minimal semi-prime submodule of M if and only if there exists a minimal semi- prime ideal A, with 𝑎𝑛𝑛 𝑀 𝐴 such that 𝐸 𝐴𝑀 𝑀. (b) Eveery semi-prime submodule of M contains a minimal semi-prime submodule.      173 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Lemma(3.3) If M is a faithful and multiplication module, and E be a non zero semi-prime submodule of M. If E is not minimal semi-prime, then E "is a weak-essential submodule of M". Proof Since M is a multiplication, and E is a semi-prime submodule of M, then by [13,prop(2.5), P.36] ∃ a "semi-prime ideal K of R" with 0 𝑎𝑛𝑛𝑀 𝐾 and E=KM. "Let S be a non-zero semi-prime submodule of M" such that 𝐸 ∩ 𝑆 0 .But E is not minimal semi-prime, then by remark(3.2)(b) every semi-prime submodule of M contain a minimal semi-prime submodule say 𝐸 𝐸. Hence by remark(3.2)(a), there exists a minimal semi-prime ideal 𝐾 of R such that 𝑎𝑛𝑛 𝑀 𝐾 and 𝐸 𝐾 𝑀 𝑀, 𝐾 ∩ 𝑆: 𝑀 𝑀 𝐾𝑀 ∩ 𝑆: 𝑀 𝑀 𝐸 ∩ 𝑆 0 0 . But M is faithful, then 𝐾 ∩ 𝑆: 𝑀 0 , which implies that 𝐾 ∩ 𝑆: 𝑀 𝐾 , that is either 𝐾 𝐾 𝑜𝑟 𝑆: 𝑀 𝐾 . If 𝐾 𝐾 , then 𝐾𝑀 𝐾 𝑀, implies that 𝐸 𝐸 which is a contradiction. Thus, 𝑆: 𝑀 𝐾 . That is 𝑆: 𝑀 𝑀 𝐾 𝑀, implies that 𝑆 𝐸 𝐸 which is contradict the minimality of 𝐸 . Thus 𝐸 ∩ 𝑆 0 is not true. Thus 𝐸 ∩ 𝑆 0 , which implies that E is a weak essential submodule of M. Proposition (3.4) If M is a faithful and multiplication module, and E be a non-zero semi-prime submodule and w-closed submodule of M, then E is a minimal semi-prime submodule of M. Proof Suppose that E is not minimal semi-prime submodule of M, then by lemma(3.3), E "is a weak essential submodule of M". But E is a w-closed submodule in M, then E=M. On the other hand E is a semi-prime submodule of M, that E must be a proper submodule of M, so we get contradiction. Hence E must be a minimal "semi-prime submodule of M". Proposition (3.5) Let M be a non zero multiplication module with only one non zero maximal submodule E. Then E can not be w-closed submodule in M. Proof Assume that E is a w-closed submodule in M, then by [3, prop(2.20)] E "is a weak essential submodule of M". Hence E=M. "But this contradict the maximality of E". Therefore E is not W-closed submodule in M. "Recal that for any module M and any ideals I and J of R if I is a semi-prime ideal of J then IM is a semi-prime submodule of JM this is called condition ∗ in [3]". Proposition(3.6) Let M be a faithful and multiplication module such that M satisfies condition ∗ , if L is a w-closed ideal in K then LM is a w-closed submodule in KM. Proof Suppose that L is a w-closed ideal in K, and LM is a weak essential submodule of T where T is a submodule of KM, we have to show that LM=T. Since M is a multiplication module, then T=PM for some ideal P of R with 𝑃 𝐾. That is LM "is a weak essential submodule of PM", and since M is faithful and satisfies condition ∗ then by [3,prop(2.17)], we have L is a weak      174 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 essential ideal in P and 𝑃 𝐾. But L is a w-closed ideal in K, then L=P. That is LM=PM=T. Hence LM is a w-closed submodule in KM. The following proposition gives the converse of proposition(3.6). Proposition (3.7) If M is a finitely generated,faithful and multiplication module, and LM is a w-closed submodule in KM, then L is a w-closed ideal in K. Proof Suppose that LM is a w-closed submodules in KM, where L and K are ideals in R, and let L is a weak essential ideal in U where U is an ideal of K. "Since M is finitely generated faithful and multiplication", then by [3, prop(2.18)] we have LM is a weak essential in UM which is a submodule of KM. But LM is a w-closed submodule in KM, then LM=UM. Hence by [12, Th,(3.1)], L=U. Then L is a w-closed ideal in K. From proposition (3.6) and proposition(3.7) we get the following corollary. Corollary(3.8) "If M is a finitely generated faithful and multiplication module which satisfies condition ∗ ", then L is a w-closed ideal in K if and only if LM is a w-closed submodule in KM. Theorem(3.9) If M is a finitely generated faithful and multiplication module, and let E be a submodule of M, such that M satisfies condition ∗ , "then the following statements are equivalent" : 1- E is a w-closed submodule in M. 2- 𝐸 : 𝑀 is a w-closed ideal in R. 3- E=PM for some w-closed ideal P in R. Proof 1 ⟹ 2 Suppose that E is a w-closed submodule in M. Since M is a multiplication, then by [7] 𝐸 𝐸 : 𝑀 𝑀. Put 𝐸 : 𝑀 𝑃, then we have PM=E is a w-closed submodule in M. Hence by cor(3.8), P is a w-closed ideal in R. That is 𝐸 : 𝑀 is a w-closed ideal in R. 2 ⟹ 3 : Suppose that 𝐸 : 𝑀 is a w-closed ideal in R. Then 𝐸 𝐸 : 𝑀 𝑀 since M is multiplication,i.e E=PM where 𝑃 𝐸 : 𝑀 is a w-closed ideal in R. 3 ⟹ 1 : Suppose that E=PM for some w-closed submodule P in R. Then by cor(3.8), PM=E is a w-closed submodule in RM=M. 4- Chain conditions on w-closed submodules We start this section by introducing the definitions of a modules that have ascending (descending) chain condition on w-closed submodules. Definition(4.1) A module M is said to have the ascending chain condition on w-closed      175 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 submodule( briefly acc on w-closed submodules ), if every ascending chain 𝐸 ⊆ 𝐸 ⊆ . . . of w-closed submodule in M is finite. That is ∃ 𝑚 ∈ 𝑍 such that 𝐸 𝐸 for all 𝑛 𝑚. Definition(4.2) A module M is said to have the descending chain condition on w-closed submodule( briefly dcc on w-closed submodules ), if every descending chain 𝐸 ⊇ 𝐸 ⊇ . . . of w-closed submodule in M is finite. That is ∃ 𝑚 ∈ 𝑍 such that 𝐸 𝐸 for all 𝑛 𝑚. Remarks (4.3) 1- 𝑍𝑝 as a Z-module satisfies dcc on w-closed submodules, but 𝑍𝑝 as a Z-modules does not satisfies acc on w-closed submodules because 𝑍𝑝 is an artinian but not noetherian. 2- Z as Z-module satisfies (acc) on w-closed submodules, but does not satisfies (dcc) on w- closed submodules because Z as a Z-module is a noetherian but not artinian. Proposition (4.4) If M is a module and satisfies (dcc) on closed submodules, then M satisfies (dcc) on w-closed submodules. Proof Let 𝐸 ⊇ 𝐸 ⊇ . . . "be a descending chain" of w-closed submodules of M. But by remark(2.2) every w-closed submodule is closed, then 𝐸 is a closed submodule for each i=1,2,. . . . Since M satisfies (dcc) on closed submodule, then ∃ 𝑚 ∈ 𝑍 such that 𝐸 𝐸 for each 𝑛 𝑚. Thus, M satisfies (dcc) on w-closed submodules. The proof of the following proposition is similar to the proof of proposition (4.4) and hence is omited. Proposition (4.5) If M is a module and satisfies (acc) on closed submodules, then M satisfies (acc) on w- closed submodules. Since w-closed submodules and closed submodules are equivalent in the class of fully semi- prime modules by proposition (2.14), "we get the following results". Proposition (4.6) If M is a fully semi-prime module, then M satisfies (acc) on w-closed submodules if and only if M satisfies (acc) on closed submodules. Proof ⟹ Let 𝐸 ⊆ 𝐸 ⊆ . . . "be ascending chain of closed submodules". Then by prop(2.14), 𝐸 is a w-closed submodule for each i=1,2, . . . . But M satisfies (acc) on w- closed submodules, so ∃ 𝑚 ∈ 𝑍 such that 𝐸 𝐸 for all 𝑛 𝑚. Thus M satisfies (acc) on closed submodules. ⟸ By proposition (4.5). The proof of the following proposition is similar to proof of proposition (4.6). Proposition (4.7) Let M be a fully semi-prime module. "Then M satisfies (dcc) on closed submodules if and only if M satisfies (dcc)" on w-closed submodules.      176 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition(4.8) If M is a module , and 𝐸 ⊆ 𝐸 ⊆ . . . "be ascending chain of submodules such that" each weak essential extension of 𝐸 is a completely essential for each i=1,2, . . . , then M satisfies (acc) on w-closed submodules if and only if M satisfies (acc) on closed submodules. Proof ⟹ Let 𝐸 ⊆ 𝐸 ⊆ . . . "be ascending chain" of closed submodules. Then by prop(2.13), 𝐸 is a w-closed submodules for each i=1,2, . . . . But M satisfies (acc) on w- closed submodules, then there exists a non zero integer m such that 𝐸 𝐸 for all 𝑛 𝑚. Hence M satisfies (acc) on closed submodules. ⟸ Follows by proposition (4.5). The proof the following proposition is similar to proof of proposition (4.8). Proposition(4.9) If M is a module, and 𝐸 ⊇ 𝐸 ⊇ . . . be a descending chain of submodules such that each weak essential extension of 𝐸 is a completely essential for each i=1,2, . . . . Then M satisfies (dcc) on w-closed submodules if and only if M satisfies (dcc) on closed submodules. Proposition(4.10) If M is a module, and D be a submodule of M such that 𝐷 𝑆𝑟𝑎𝑑 𝑀 ∩ 𝐾, where K is any w-closed submodule in M. If 𝑴 𝑫 satisfies (dcc) on w-closed submodules, then M satisfies (dcc) on w-closed submodules. Proof Let 𝐸 ⊇ 𝐸 ⊇ . . . be a descending chain of w-closed submodules in M. Since 𝐸 is a w-closed submodule in M for each i=1,2, . . . , and 𝐷 𝑆𝑟𝑎𝑑 𝑀 ∩ 𝐸 then by Corollary(2.32), we have 𝑬𝒊 𝑫 is a w-closed submodule in 𝑴 𝑫 for each i=1,2, . . . . Hence ⊇ ⊇ . . . , is a descending chain of w-closed submodules in 𝑴 𝑫 . But 𝑴 𝑫 satisfies (dcc) on w-closed submodules, so there exists a positive integer m such that 𝑬𝒏 𝑫 𝑬𝒎 𝑫 for each 𝑛 𝑚. So, that 𝐸 𝐸 for each 𝑛 𝑚. Thus M satisfies (dcc) on w-closed submodules. Proposition(4.11) If M is a module, and D be a submodule of M such that 𝐷 𝑆𝑟𝑎𝑑 𝑀 ∩ 𝐾, where K is any w-closed submodule in M. If 𝑴 𝑫 satisfies (acc) on w-closed submodules, then M satisfies (acc) on w-closed submodules. Proof Similar to proof of proposition (4.10). Proposition(4.12) If 𝑋 𝑋 ⨁ 𝑋 is a module, where 𝑋 and 𝑋 are submodules of X, provided that ann 𝑋 + ann 𝑋 =R, and all weak essential extensions of 𝐸 ⨁ 𝑋 (or 𝑋 ⨁ 𝐸      177 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 ) are completely essential modules where 𝐸 is a non zero w-closed submodule in 𝑋 (or 𝑋 ) for each i=1,2, . . . . If X satisfies (dcc) on w-closed submodules, then 𝑋 (𝑜𝑟 𝑋 ) satisfies (dcc ) on non- zero w-closed submodules. Proof Let 𝐸 ⊇ 𝐸 ⊇ . . . "be a descending chain" of a non-zero w-closed submodules of 𝑋 . If 𝑋 is equal to zero, then X=𝑋 and this, implies that 𝑋 satisfies (dcc) on non-zero w-closed submodules. Otherwise, since 𝐸 is a non-zero w-closed submodule in 𝑋 , and 𝑋 is a w- closed in 𝑋 , so by proposition(2.26), 𝐸 ⨁ 𝑋 is a w-closed submodule in X for each i=1,2, . . . , 𝐸 ⨁𝑋 ⊇ 𝐸 ⨁𝑋 ⊇. . . , "is a descending chain" of w-closed submodule in X. But X satisfies (dcc) on w-closed submodules, then there exists a positive integer m such that 𝐸 ⨁𝑋 𝐸 ⨁𝑋 for all 𝑛 𝑚. Thus 𝐸 𝐸 for all 𝑛 𝑚. Thus 𝑋 satisfies (dcc) on w-closed submodule. Similarly we can prove that 𝑋 satisfies (dcc) on w-closed submodule. Proposition(4.13) If 𝑋 𝑋 ⨁ 𝑋 is a module, where 𝑋 and 𝑋 are submodules of X, provided that ann 𝑋 + ann 𝑋 =R, and all weak essential extensions of 𝐸 ⨁ 𝑋 (or 𝑋 ⨁ 𝐸 ) are completely essential modules where 𝐸 is a non zero w-closed submodule in 𝑋 (or 𝑋 ) for each i=1,2, . . . . If X satisfies (acc) on w-closed submodules, then 𝑋 (𝑜𝑟 𝑋 ) satisfies (acc ) on non- zero w-closed submodules. Proof Similarly as in proposition (4.12). We end this section by the following propositions. Proposition(4.14) "If M is a finitely generated,faithful and multiplication module, and M satisfies condition ∗ ", then M satisfies (dcc) on w-closed submodules, if and only if R satisfies (dcc) on w-closed ideals. Proof ⟹ Let 𝐿 ⊇ 𝐿 ⊇ . . . , "be a descending chain" of w-closed ideals in R. Since 𝐿 is a w-closed ideal in R for each i=1,2, . . . . Then by cor(3.8) 𝐿 𝑀 is a w-closed submodule in M for each i=1,2, . . . , then 𝐿 𝑀 ⊇ 𝐿 𝑀 ⊇ . . . , be a "descending chain" of w-closed submodules in M. But M satisfies (dcc) on w-closed submodules, "so there exists a positive integer m such that" 𝐿 𝑀 𝐿 𝑀 for each 𝑛 𝑚. But M is a finitely generated faithful and multiplication R-module, then by [12, Th(3.1)], 𝐿 𝐿 foe each 𝑛 𝑚. Therefore R satisfies (dcc) on w-closed ideals. ⟸ Let 𝐸 ⊇ 𝐸 ⊇ . . . , be a descending chain of w-closed submodules in M. Since M is multiplication module, then 𝐸 𝐿 𝑀 for some ideal 𝐿 of R ∀ i=1,2, . . . , then 𝐿 𝑀 ⊇ 𝐿 𝑀 ⊇ . . . . Since 𝐸 is a w-closed submodule in M for each i=1,2, . . . , so by cor(3.8), 𝐿 is a w-closed ideal in R for each i=1,2, . . . , . But M is a finitely generated, faithful and multiplication module, then by [12, Th(3.1)] we have 𝐿 ⊇ 𝐿 ⊇ . . . , is a "descending chain" of w-closed ideals in R. But R satisfies (dcc) on w-closed ideals, therefore, there exists a positive integer m such that 𝐿 𝑀 𝐿 𝑀 for each 𝑛 𝑚, thus 𝐸 𝐸 for each 𝑛 𝑚. The proof the following proposition is similar to the proof of prop(4.14), hence we omited.      178 |  Mathmatics 5https://doi.org/10.30526/31.2.195  2018) عام 2العدد( 31لمجلد ا مجلة إبن الهيثم للعلوم الصرفة و التطبيقية Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 31 (2) 2018 Proposition(4.15) "If M is a finitely generated,faithful and multiplication module, and M satisfies condition ∗ ", then M satisfies (acc) on w-closed submodules, if and only if R satisfies (acc) on w-closed ideals. References 1. GoodearL K.R. (1976)., " Ring Theory, Nonsingular Rings and Modules ", Marcel, Dekker, New York, 2. Muna ,A. A. (2009) , " Weak Essential Submodules ", Um-Salma Science Journal, Vol. 6(1).214-222. 3. Muna, A. A. (2016) " Some results on weak Essential submodules ", Baghdad Science Journal to apper. 4. Dauns, J. and B. R. McDonald, (1980), " Prime modules, and one-sided ideals in Rings and Algebra III ", Proceeding of third Oklahoma, Conference 301-344. 5. Athab E. 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