Microsoft Word - 353-362 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|353    Strongly 𝑪𝟏𝟏-Condition Modules and Strongly 𝑻𝟏𝟏-Type Modules Inaam M A Hadi Inaam1976@yahoo.com Dept. of Mathematics, College of Education for Pure Science (Ibn-Al-Haitham) University of Baghdad Farhan D Shyaa farhan.shyaa@qu.edu.iq Dept. of Mathematics, University of Al-Qadsiyah Abstract In this paper, we introduced module that satisfying strongly 𝐶 -condition modules and strongly 𝑇 -type modules as generalizations of t-extending. A module 𝑀 is said strongly 𝐶 - condition if for every submodule of 𝑀 has a complement which is fully invariant direct summand. A module 𝑀 is said to be strongly 𝑇 -type modules if every t-closed submodule has a complement which is a fully invariant direct summand. Many characterizations for modules with strongly 𝐶 -condition for strongly 𝑇 -type module are given. Also many connections between these types of module and some related types of modules are investigated. Keywords. 𝐶 -condition, strongly 𝐶 -condition modules, 𝑇 -type modules strongly 𝑇 -type modules, t-semisimple modules, strongly t-semisimple modules, strongly extending modules, t-extending modules and strongly t-extending modules. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|354    1. Introduction First, a few concepts and some results which are relevant for our work are recalled. Throughout, all rings are associative rings with unity and all modules are right unitary modules. A submodule 𝑁 of 𝑀 is called closed in 𝑀 if has no proper essential extension in 𝑀[8], that means if 𝑁 is essential in 𝑊, where 𝑊 𝑀,then 𝑁 𝑊, where a submodule 𝑁 is essential in an 𝑅-module 𝑀 (briefly (𝑁 𝑀 if 𝑁 ∩ 𝐾 0 , 𝐾 𝑀 implies 𝐾 0 [8] . As a generalization of essential submodule, Asgari in [2], introduced the notion of t- essential submodule, where a submodule 𝑁 of 𝑀 is called t-essential (denoted by 𝑁 𝑀 if whenever 𝑊 𝑀, 𝑁 ∩ 𝑊 𝑍 𝑀 implies 𝑊 𝑍 𝑀 . 𝑍 𝑀 is called the second singular submodule and is defined by 𝑍 [8], where 𝑍 𝑀 𝑥 ∈ 𝑀: 𝑥𝐼 0 for some essential ideal of 𝑅}. Equivalently 𝑍 𝑀 𝑥 ∈ 𝑀: 𝑎𝑛𝑛 𝑥 𝑅 and 𝑎𝑛𝑛 𝑥 𝑟 ∈ 𝑀: 𝑥𝑟 0 . 𝑀 is called singular (nonsingular) if 𝑍 𝑀 𝑀 𝑍 𝑀 0 . Note that 𝑍 𝑀 𝑥 ∈ 𝑀: 𝑥𝐼 0 for some t-essential ideal 𝐼 of 𝑅 .𝑀 is called 𝑍 -torsion if 𝑍 𝑀 𝑀[8]. A submodule 𝑁 is called t-closed (denoted by 𝑁 𝑀 if 𝑁 has no proper t-essential extension in 𝑀[2]. It is clear that every t-closed submodule is closed, but the convers is not true. However, under the class of nonsingular, the two concepts are equivalent. Recall that  a module 𝑀 is called extending if for every submodule 𝑁 of 𝑀 then there exists a direct summand 𝑊 𝑊 ⨁ 𝑀) such that 𝑁 𝑊  [6]. Equivalently 𝑀 is extending module if every closed submodule of 𝑀 is a direct summand. As a generalization of extending module, Asgari [2] introduced the concept t-extending module, where a module 𝑀 is t-extending if every t-closed submodule is a direct summand. Equivalently,𝑀 is t-extending if every submodule of 𝑀 is t-essential in a direct summand [2]. The notion of a strongly extending module is introduced in [13], which is a subclass of the class of extending module, where an R-module M is called strongly extending if each submodule of M is essential in a fully invariant direct summand of M , where a submodule N is called fully invariant if for each 𝑓 ∈ End M , 𝑓 N N.An 𝑅-module is called strongly t-extending if every submodule 𝑁 of M, there exists a fully invariant direct summand 𝑊 of M such that 𝑁 𝑊[7].Equivalently “M is strongly t-extending if every t-closed submodule of 𝑀 is fully invariant direct summand[7]. A module M is called duo if every submodule of M is fully invariant [12]. Hence the two concepts strongly extending and extending are equivalent in the class of duo modules. Asgari and Haghany introduced the concept of t-semisimple modules and t-semisimple rings. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t- essential in N [3]. In [9] Inaam and Farhan introduced and studied strongly t-semisimple, where an R-module is called strongly t-semisimple if for each submodule N of M there exists a fully invariant direct summand K such that K N [9]. Inaam and Farhan in [10] introduced and studied FI-t-semisimple and strongly FI-t-semisimple. An R-module M is called FI-t- semisimple if for each fully invariant submodule 𝑁 of 𝑀, there exists K ⨁ M such that K N. An R-module 𝑀 is called strongly FI-t-semisimple if for each fully invariant submodule 𝑁 of 𝑀, there exists a fully invariant direct summand 𝐾 such that K N [10]. Recall that: An 𝑅-module 𝑀 is said to be satisfy 𝐶 -condition if every submodule of 𝑀 has a complement which is a direct summand[15]. Asgari [4], restricted 𝐶 condition to t- closed condition of 𝑀. She defined the following. An 𝑅-module 𝑀 said to be 𝑇 -type module (or 𝑀 satidfy 𝑇 -condition) if every t-closed submodule has a complement which is a direct summand. A ring is said to be right 𝑇 -type ring if 𝑅 is a 𝑇 -type module [4]. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|355    This paper consists of three sections. In section two we deal with certain known results which are worthwhile throughout the paper. In section three we investigate certain types of module namely strongly 𝐶 -condition and strongly 𝑇 -type modules. An 𝑅-module 𝑀 is said to be satisfy strongly- 𝐶 -condition if every submodule of 𝑀 has a complement which is a fully invariant direct summand. Also; an 𝑅-module M is called strongly 𝑇 -type module if every t- closed submodule has a complement which is a fully invariant direct summand. We give several characterizations of strongly 𝐶 -condition and strongly 𝑇 -type module. In particular 𝑀 is strongly 𝑇 -type module if and only if 𝑀 𝑍 𝑀 ⨁𝑀 , where 𝑀 satisfies strongly 𝐶 -condition. Every module with strongly 𝐶 -condition is strongly 𝑇 -type module, but not conversely (see Remark 3.6(1)). The two concepts are equivalent under certain classes of module is given. Furthermore, many connections between strongly 𝑇 -type modules strongly t- semisimple module, strongly FI-t-semisimple module, FI-t-semisimple module, strongly extending module, strongly t-extending module are presented. 2. Preliminaries Proposition (1.1)[2]: The following statements are equivalent for a submodule 𝐴 of an 𝑅- module 𝑀 𝐴 is t-essential in 𝑀 ; (𝐴+𝑍 𝑀 /𝑍 𝑀 is essential in 𝑀/𝑍 𝑀 ; 𝐴+𝑍 𝑀 is essential in 𝑀; 𝑀/𝐴 is 𝑍 𝑡𝑜𝑟𝑠𝑖𝑜𝑛.. Lemma (1.2)[2]: Let M be an 𝑅- module. Then If M, then 𝑍 𝑀 𝐶. 0 𝑀 if and only if M is nonsingular. If 𝐴 𝐶, then 𝐶 M if and only if . Theorem (1.3)[9]:  The following statements are equivalent for an 𝑅 -module 𝑀: 𝑀 is strongly t-semisimple, is a fully stable semisimple and isomorphic to a stable submodule of 𝑀, 𝑀 = 𝑍 (𝑀)⨁𝑀 where 𝑀 is a nonsingular semisimple fully stable module and 𝑀 is stable in 𝑀, Every nonsingular submodule is stable direct summand, Every submodule of 𝑀 which contains 𝑍 (𝑀) is a direct summand of 𝑀 and is fully stable and isomorphic to a stable submodule of 𝑀. Proposition (1.4)[10]: Let 𝑀 be an 𝑅-module with the property, complement of any submodule of 𝑀 is stable. The following statements are equivalent. 𝑀 is strongly FI-t-semisimple; 𝑀 is FI-t-semisimple;. Let (⋇) means the following: For an 𝑹-module 𝑴, the complement of 𝒁𝟐 𝑴 is stable in 𝑴[10]. Proposition (1.5)[10]: Let 𝑀 be an 𝑅-module which satisfies (⋇ . If 𝑀 is strongly FI-t- semisimple, then is FI-semisimple, and hence it is strongly FI-t-semisimple. Corollary (1.6)[10]: For an 𝑅-module 𝑀 which satisfies ⋇ 𝑀 is strongly FI-t-semisimple if and only if for every fully invariant submodule 𝑁 of 𝑀 such that 𝑁 ⊇ 𝑍 𝑀 , is strongly FI-t- semisimple,. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|356    Proposition (1.7)[2]: Let 𝑀 be a nonsingular module. Then 𝑀 is strongly extending if and only if 𝑀 is strongly t-extending. 2. T-semisimple modules and 𝑻𝟏𝟏-type modules Remarks and Examples (2.1): It is clear that module satisfying 𝐶 implies 𝑇 -type-module [4]. Every t-extending module (hence every extending module) is a 𝑇 -type module [4]. Every 𝑍 -torsion is 𝑇 -type module and every finite generated abelian group is a 𝑇 -type module [4]. Proposition (2.2): Every t-semisimple module is 𝑇 -type module. Proof: By [3, Proposition 2.16], every t-semisimple is t-extending, hence by Remarks and Examples 2.1(2) it is 𝑇 -type module. □ Remark (2.3)[4]: The class of 𝑇 -type modules properly contains the modules satisfying 𝐶 condition and the class of t-extending modules . Examples (2.4): Let 𝑅 𝑍 𝑋 , 𝑅 is uniform, nonsingular 𝑅-module. By [13, Theorem 2.4] 𝑅⨁𝑅 satisfies 𝐶 -condition. Hence 𝑅⨁𝑅 is 𝑇 -type module. But 𝑅⨁𝑅 is not t-semisimple ,because it is so, then 𝑅⨁𝑅 is t-extending, which is a contradiction since by [5,Example 2.4] 𝑅⨁𝑅 is not extending, hence not t-extending, since 𝑅⨁𝑅 is nonsingular see Remarks and Examples 2.1(2). The 𝑍-module 𝑍 is not t-semisimple. But 𝑍 is indecomposable and nonsingular uniform, so 𝑍 is 𝑇 -type module see[4, Corollary 2.8]. An 𝑍-module 𝑄 is indecomposable, nonsingular, uniform so 𝑄 is 𝑇 -type module[4, Corollary 2.8] Any direct summand of uniform is 𝐶 -type module, so is 𝑇 -type module [15]. (5) it is clear the 𝑍 module 𝑄⨁𝑍, 𝑍 ⨁𝑍 , 𝑍 ⨁𝑍 are 𝑇 -type module. Also notice that 𝑄⨁𝑍 is not t-semisimple. 3. Strongly 𝑪𝟏𝟏-modules and strongly 𝑻𝟏𝟏-type modules. In this section, we generalize modules with 𝐶 -condition and 𝑇 -type modules into modules with strongly 𝐶 - conditions and strongly 𝑇 -type modules. We study these concepts and their connection with strongly t-semisimple modules and other related classes of modules. Definition (3.1): An 𝑅-module 𝑀 said to be satisfy strongly 𝐶 - condition (𝑀 is strongly 𝐶 ) if every submodule has a complement which is fully invariant direct summand. Lemma (3.2)[15]: Let 𝑁 𝑀, let 𝐾 be a direct summand of 𝑀 . 𝐾 is a complement of 𝑁 if and only if 𝐾⋂𝑁 0 and 𝐾⨁𝑁 𝑀 . The following Lemma is clear. Lemma (3.3): If 𝑁 𝑀 and 𝐾 is a fully invariant direct summand then 𝐾 is a fully invariant complement of 𝑁 if and only if 𝐾⋂𝑁 0 and 𝐾⨁𝑁 𝑀. The following Proposition gives characterizations for module with strongly 𝐶 -condition. Proposition (3.4): The following statements are equivalent for a module 𝑀 𝑀 satisfies strongly 𝐶 -condition; For any complement submodule 𝐿 in 𝑀, there exists a fully invariant direct summand 𝐾 of 𝑀 such that 𝐾 is a complement of 𝐿 in 𝑀; For any submodule 𝑁 of 𝑀, there exists a fully invariant direct summand 𝐾 of 𝑀 such that 𝑁⋂𝐾 0 and 𝑁⨁𝐾 is an essential submodule of 𝑀; For any complement submodule 𝐿 in 𝑀, there exists a fully invariant direct summand 𝐾 of 𝑀 such that 𝐿⋂𝐾 0 and 𝐿⨁𝐾 𝑀. Proof: (1)  (2) For any complement submodule 𝐿 in 𝑀. By strongly 𝐶 -condition, there exists a fully invariant direct summand 𝐾 of 𝑀 which is a complement of 𝐿 in 𝑀. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|357    (3)  (4) and (2)  (4) are obvious. (1)  (3) It is clear by Lemma (3.3). (4)  (1) Let 𝐴 be any submodule of 𝑀. Then there exists a complement (so closed submodule 𝐵 in 𝑀) such that 𝐴 𝐵, by [8,Excerces 13,P.20]. By hypothesis, there exists a fully invariant direct summand 𝐾 of 𝑀 such that 𝐵⋂𝐾 0 and 𝐵⨁𝐾 𝑀. Hence by Lemma (3.2) 𝐾 is a complement of 𝐵 in 𝑀. So 𝐵⋂𝐾 0, which implies 𝐾⋂𝐴 0. Suppose that 𝐾 𝑀 and, 𝐾 𝐾. Therefore 𝐾 ⋂𝐵 0 and hence 𝐾 ⋂𝐵 ⋂𝐴 0 (since 𝐴 𝐵 , so that 𝐾 ⋂𝐴 0. Thus 𝐾 is a complement of 𝐴 in 𝑀. □ As we have seen t-semisimple is 𝑇 -type module. We claim that strongly t-semisimple modules imply modules which are strongly than module with 𝑇 -type module. Hence this leads us to define the following: Definition (3.5): An 𝑅-module is said to be strongly 𝑇 (or strongly 𝑇 -type modules) if for each t-closed submodule, there exists a complement which is a fully invariant direct summand. Remarks (3.6): (1) It is clear that every module, which satisfies strongly 𝐶 -condition, is a strongly 𝑇 -type module, but the converse is not true in general, as the following example shows: Let 𝑀 𝑍 ⨁𝑍 as 𝑍-module 𝑀 is 𝑇 -type module by [4, Corollary 2.6]. 𝑀 is strongly 𝑇 -type module, but it is not strongly 𝐶 -type module. If 𝑁 2 ⨁ 0 2, 0 , 4, 0 , 6, 0 , 0, 0 , 𝑁 ∩ 0 ⨁𝑍 0, 0 and 𝑁⨁𝑊 2)⨁𝑍 𝑀 where 𝑊 0 ⨁𝑍 and 𝑊 ⨁ 𝑀, then 𝑊 is a complement of 𝑁. Also 𝑁 ∩ 𝐾 0 , where 𝐾 (4, 1 , 0, 0)}. But ⨁𝐾 0, 0 , 2, 0),( 4, 0),( 6, 0)}⨁ 4, 1 , 0, 0 = 4, 1 , 0, 0 , 6, 1 , 2, 0 , 0, 1 , 4, 0),( 2, 1 , 6, 0 𝑈 and 𝑈 𝑀, hence is a complement of 𝑁, but 𝐾 ≰⨁ 𝑀. Thus 𝑊 is a unique complement of 𝑁 which is a direct summand, but 𝑊 is not fully invariant submodule as there exists 𝑓: 𝑊 0, 0 , 0, 1 ⟼ 𝑀 defined by 𝑓 0, 0 0, 0 𝑓 0, 1 4, 1 , 𝑓 is 𝑍-homorphism and 𝑓 𝑊 ≰ 𝑊. Thus 𝑀 does not satisfy strongly 𝐶 -type module. Also 𝑀 is singular, so 𝑀 is the only t-closed submodule and has a complement which is the zero submodules and it is clear direct summand fully invariant submodule. Thus 𝑀 is strongly 𝑇 -type. (2) Let 𝑀 be an 𝑅-module which satisfies strongly 𝐶 -condition. Then 𝑀 is strongly FI-t- semisimple if and only if 𝑀 is FI-t-semisimple. Proof: It follows directly by Proposition 1.4, and Definition 3.1. □ Proposition (3.7): Let 𝑀 be a nonsingular 𝑅-module. 𝑀 is strongly 𝐶 -condition module if and only if 𝑀 is strongly 𝑇 -type module. Proof: It is clear.  Let 𝐴 𝑀, by [8,Exercies 13,P.20], there exists a closed submodule 𝑊 of 𝑀 such that 𝐴 𝑊. Since 𝑀 is nonsingular, 𝑊 is t-closed in 𝑀. Hence there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑊⨁𝐷 𝑀. It follows that 𝐴⨁𝐷 𝑊⨁𝐷 𝑀. Thus 𝐴⨁𝐷 𝑀. So that 𝑀 satisfies strongly 𝐶 -condition. □ Recall that an 𝑅-module 𝑀 is called multiplication module if for each 𝑁 𝑀, there exists an ideal 𝐼 of 𝑅 with 𝑁 𝑀𝐼[17]. Equivalently an 𝑅-module 𝑀 is a multiplication module if for each 𝑁 𝑀,𝑁 𝑁: 𝑀 𝑀, where 𝑁: 𝑀 𝑟 ∈ 𝑅: 𝑀𝑟 𝑁 ".[17] Proposition (3.8): Let 𝑀 be a multiplication (hence 𝑀 is duo or fully stable). Then 𝑀 is 𝑇 -type module if and only if 𝑀 is strongly 𝑇 -type module. 𝑀 is 𝐶 -type if and only if 𝑀 is strongly 𝐶 -type module. We will give some properties of strongly 𝑇 -type modules. Theorem (3.9): Consider the following statements for a module 𝑀 𝑀 is strongly 𝑇 -type module; IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|358    𝑀 𝑍 𝑀 ⨁𝑀 , where 𝑀 is a fully invariant submodule in 𝑀 and satisfies strongly 𝐶 - condition; For every submodule 𝐴 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐴⨁𝐷 𝑀. For every t-closed submodule 𝐶 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐶⨁𝐷 𝑀. For every t-closed submodule 𝐶 of 𝑀, there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐶⨁𝐷 𝑀. Then (1), (3), (4) and (5) are equivalent, (1) (2) if is fully invariant of for each fully invariant submodule 𝐿 of 𝑀 containg 𝑍 𝑀 , and (2)  (5). Proof: (1)(5) Let 𝐶 be a t-closed submodule of 𝑀. By condition (1) there exists a complement 𝐷 to 𝐶 such that 𝐷 ⨁ 𝑀, 𝐷 is fully invariant. Thus 𝐶⨁𝐷 𝑀. (3) (1) Let 𝐶 be a t-closed submodule of 𝑀. By hypothesis there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐶⨁𝐷 𝑀. Let 𝐸 be a complement of 𝐶, then 𝐶⋂𝐸 0 and 𝐶⨁𝐸 𝑀. We claim that 𝐶⨁𝐷 𝐶⨁𝐸. Let 𝐶⨁𝐷 ⋂𝑋 0 , where 𝑋 𝐶⨁𝐸. 𝐶⨁𝐷 ⋂𝑋 0 𝑍 𝑀 . Thus implies 𝑋 𝑍 𝑀 since 𝐶⨁𝐷 𝑀. But 𝑍 𝑀 𝐶 (since 𝐶 is t-closed) hence 𝑋 𝐶. It follows that 𝐶⨁𝐷 ⋂𝑋 𝑋 0 . Thus 𝐶⨁𝐷 𝐶⨁𝐸. It follows that 𝐷 𝐸. However,𝐷 ⨁ 𝑀 and so 𝐷 is closed in 𝑀. which implies 𝐷 𝐸,that is 𝐸 a complement of 𝐶, which is a fully invariant direct summand. Thus 𝑀 is a strongly 𝑇 -type module. (4)  (3) Let 𝐴 𝑀. By [4, Lemma 2.3], there exists a t-closed 𝐶 of 𝐴 such that 𝐴 𝐶. By hypothesis, there exists a fully invariant direct summand 𝐷 such that 𝐶⨁𝐷 𝑀. But 𝐴 𝐶, we conclude that 𝐴⨁𝐷 𝐶⨁𝐷 and hence 𝐴⨁𝐷 𝑀. (5) (4) The implication is clear since every essential submodule is t-essential submodule. (2) (5) Let 𝐶 be a t-closed submodule of 𝑀. Hence by Lemma (1.2), 𝑍 𝑀 𝐶 and so 𝐶 𝑍 𝑀 ⨁ 𝐶⋂𝑀 . Moreover, 𝐶⋂𝑀 is a t-closed submodule of 𝑀 by [2, proposition 2.9]. Since 𝑀 satisfies strongly 𝐶 condition, there exists a fully invariant direct summand 𝐷 of 𝑀 such that ( 𝐶⋂𝑀 ⨁𝐷 𝑀 . But 𝐷 ⨁ 𝑀 and 𝑀 ⨁ 𝑀, then 𝐷 ⨁ 𝑀 and 𝐶⨁𝐷 𝑍 𝑀 ⨁ 𝐶⋂𝑀 ⨁𝐷 𝑍 𝑀 ⨁ 𝐶⋂𝑀 ⨁𝐷] 𝑍 𝑀 ⨁𝑀 𝑀.Hence 𝐶⨁𝐷 𝑀, but 𝐷 is fully invariant in 𝑀 and 𝑀 is fully invariant in 𝑀. Hence 𝐷 is fully invariant in 𝑀. (1) (2) Since 𝑀 is strongly 𝑇 -type module and 𝑍 𝑀 is a t-closed submodule of 𝑀, there exists a complement 𝑀 to 𝑍 𝑀 which is a fully invariant direct summand, say 𝑀 𝐿⨁𝑀 . Since 𝑀 is nonsingular, we have 𝑍 𝑀 𝑍 𝐿 . But 𝑍 𝑀 ⨁𝑀 𝑀 since 𝑀 is complement to 𝑍 𝑀 , so by Proposition (1.1) is 𝑍 -torsion, thus 𝐿 is 𝑍 -torsion (since 𝐿 ≃ . So 𝐿 𝑍 𝐿 𝑍 𝑀 and hence 𝐿 𝑍 𝑀 . Therefore 𝑀 𝑍 𝑀 ⨁𝑀 . Now to show that 𝑀 ≃ ≃ 𝑀 satisfies strongly 𝐶 condition. Let �̅�= be a closed submodule of 𝑀 so �̅� is t-closed in 𝑀 and 𝐶 is t-closed submodule of 𝑀 by Lemma 1.2(3).But 𝑀 is a strongly 𝑇 -type, so there exists a complement 𝐷 of 𝐶 in 𝑀 which is a fully invariant direct summand of 𝑀. Say 𝑀 𝐷⨁𝐷 for some𝐷 𝑀. Since 𝑍 𝑀 𝑍 𝐷 ⨁𝑍 𝐷 we get𝑀 ⨁ ⨁ ≃ ⨁ 𝐷⨁𝐷 . Cleary 𝐷 ∩ 𝐷 =0 and �̅�⨁𝐷 𝑀. But 𝐷, 𝑍 𝑀 are fully invariant in 𝑀 and by hypothesis 𝐷 is fully invariant in 𝑀. □ IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|359    Recall that. A submodule 𝑁 of 𝑅-module 𝑀 is called stable, if 𝑓 𝑁 𝑁 for each 𝑅- homomorphism 𝑓: 𝑁𝑀 . An 𝑅-module 𝑀 is fully stable if every submodule of 𝑀 is stable.[1] Remarks (3.10): If an 𝑅-module 𝑀 is fully stable (where an 𝑅-module 𝑀 is fully stable if every submodule of 𝑀 is stable) and semisimple, then 𝑀 is strongly 𝐶 -condition module. Proof: Let 𝑁 𝑀, then 𝑁 ⨁ 𝑀, and so there exists 𝑊 𝑀 such that 𝑁⨁𝑊 𝑀, hence 𝑊 is a complement of 𝑁. But 𝑀 is fully stable, so 𝑊 is a fully invariant, moreover 𝑊 ⨁ 𝑀. Thus 𝑀 is strongly 𝐶 -condition module. □ Let 𝑀 be a strongly 𝑇 -type module. Then 𝑀 is strongly-FI-t-semisimple if and only if every fully invariant t-closed submodule of 𝑀 is strongly-FI-t-semisimple. Proof: Since 𝑀 is strongly 𝑇 -type module and 𝑍 𝑀 is t-closed, then there exists a 𝑍 𝑀 has a complement which is a fully invariant direct summand, that is 𝑍 𝑀 has a complement which is stable direct summand (so condition ∗ hold). Then by Corollary 1.6 every fully invariant submodule 𝑁 of 𝑀, 𝑁 ⊇ 𝑍 𝑀 is strongly FI-t-semisimple. Thus every fully invariant t-closed submodule of 𝑀 is strongly FI-t-semisimple. □ Let 𝑀 be a strongly 𝑇 -type module. If 𝑀 strongly FI-t-semisimple, then is FI- semisimple and hence it is strongly FI-semisimple. Proof: Since 𝑀 is strongly 𝑇 -type and 𝑍 𝑀 is t-closed, then there exists a complement of 𝑍 𝑀 which is a fully invariant direct summand. Thus (condition (∗ hold). Hence the result is followed by Proposition 1.5. □ Proposition (3.11): If an 𝑅-module 𝑀 strongly t-semisimple, then 𝑀 strongly 𝑇 -type module. Proof: By Theorem 1.3, 𝑀 𝑍 𝑀 ⨁𝑀 , where 𝑀 is nonsingular semisimple fully stable and 𝑀 is stable in 𝑀 . But 𝑀 is fully stable semisimple then 𝑀 is strongly 𝐶 -condition module by Remarks 3.10(1). Hence 𝑀 satisfies condition (2) of Theorem 3.9. where (251). Thus 𝑀 is strongly 𝑇 -type module. □ Theorem (3.12): Every strongly extending module is strongly 𝑇 -type module. Proof: Let 𝑁 be a t-closed submodule of 𝑀. Hence 𝑁 is a closed submodule. As 𝑀 is strongly extending, 𝑁 is a fully invariant direct summand. Then 𝑀 𝑁 ⨁𝑊 for some 𝑊 𝑀 and so 𝑊 is a complement of 𝑁. To see this let 𝑊 𝑀 and 𝑊 𝑊 𝑀 and 𝑁⋂𝑊 0 , then 𝑀 𝑁⨁𝑊 ⊆ 𝑁⨁𝑊 , so 𝑀 𝑁⨁𝑊 𝑁⨁𝑊. Assume 𝑥 ∈ 𝑊 then 𝑥 𝑛 𝑦, 𝑛 ∈ 𝑁, 𝑦 ∈ 𝑊 𝑊 , then 𝑥 𝑦 𝑛 ∈ 𝑁⋂𝑊 0, hence 𝑥 𝑦 0 implies 𝑥 𝑦 ∈ 𝑊. Hence 𝑊 𝑊, moreover 𝑊 ⨁ 𝑀, so 𝑊 is closed submodule and hence 𝑊 is a fully invariant direct summand. Thus 𝑀 is strongly 𝑇 -type module. □ Proposition (3.13): If 𝑀 is a strongly t-extending 𝑅-module then is strongly 𝑇 -type module and every complement to a nonsingular direct summand is fully invariant direct summand. Proof: Since 𝑀 is strongly t-extending, then 𝑀 𝑍 𝑀 ⨁𝑀 , 𝑀 is strongly extending module[7]. Hence, 𝑀 is strongly 𝑇 -type module by Proposition (3.12). But 𝑀 is nonsingular, so 𝑀 satisfies strongly 𝐶 -condition module by Proposition (3.7). Thus 𝑀 satisfies condition (2) of Theorem 3.9 so 𝑀 is a strongly 𝑇 -type module. Now let 𝐶 be a complement of a nonsingular submodule of 𝑀, so by [2, Proposition 2.6(52)] 𝐶 is a t-closed submodule of 𝑀. Hence 𝐶 is a fully invariant direct summand of 𝑀 by definition of strongly t- extending Proposition (1.7). Not that if every complement of nonsingular submodule of an 𝑅-module is fully invariant direct summand implies 𝑀 is strongly t-extending, since by [2, Proposition 2.6(52)] every t-closed IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|360    is a complement of nonsingular submodule and so that every t-closed submodule is fully invariant direct summand (that is 𝑀 is strongly t-extending). □ Proposition (3.14): Let 𝑀 𝑀 ⨁𝑀 , 𝑀 is a fully invariant submodule in 𝑀. Then the following conditions are equivalent: 𝑀 is strongly 𝑇 -type module; For every submodule 𝐴 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑀 𝐷 and 𝐴⨁𝐷 𝑀. For every t-closed submodule 𝐶 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑀 𝐷 and 𝐶⨁𝐷 𝑀; For every t-closed submodule 𝐶 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑀 𝐷 and 𝐶⨁𝐷 𝑀. Proof: (1)  (2) Since 𝑀 is strongly 𝑇 -type module, then by condition (3) of Theorem 3.9 for each 𝐴 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐴⨁𝐷 𝑀 . But 𝐷 ⨁ 𝑀 implies that 𝐷⨁𝑀 ⨁ 𝑀. Also, we can show that 𝐷⨁𝑀 is fully invariant in 𝑀. Let 𝑓 ∈ 𝐸𝑛𝑑 𝑀 𝐸𝑛𝑑 𝑀 𝐻𝑜𝑚 𝑀 , 𝑀 𝐻𝑜𝑚 𝑀 , 𝑀 𝐸𝑛𝑑 𝑀 . But 𝑀 is fully invariant in 𝑀 by hypothesis so Hom(𝑀 , 𝑀 0 , then 𝑓 𝑓 0 𝑓 𝑓 for some 𝑓 ∈ 𝐸𝑛𝑑 𝑀 , 𝑓 ∈ 𝐻𝑜𝑚 𝑀 , 𝑀 , 𝑓 ∈ 𝐸𝑛𝑑 𝑀 .Hence 𝑓 𝐷⨁𝑀 ≃ 𝑓 0 𝑓 𝑓 𝐷 𝑀 𝐷 𝑀 , hence 𝐷⨁𝑀 is fully invariant in 𝑀. Moreover, 𝐴⨁𝐷 𝑀 implies that 𝐴⨁𝐷 ⨁𝑀 𝑀 ⨁𝑀 𝑀. (2)  (3) It is obvious (3)  (4) For every t-closed submodule 𝐶 of 𝑀 , there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑀 𝐷 and 𝐶⨁𝐷 𝑀 . Then 𝐶⨁𝐷 𝑍 𝑀 𝑀 by Proposition (1.1). But 𝑍 𝑀 𝑍 𝑀 ⨁𝑍 𝑀 . As 𝐶 is t-closed in 𝑀 ,𝐶 ⊇ 𝑍 𝑀 by Lemma 1.2(1). Also as 𝑀 𝐷, then 𝑍 𝑀 𝑍 𝐷 𝐷. It follows that ⨁𝐷 𝑍 𝑀 𝐶⨁𝐷 𝑍 𝑀 ⨁𝑍 𝑀 𝐶⨁𝐷 𝑀. (4)  (1) Let 𝐶 be a t-closed of 𝑀 . By condition (4) there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝑀 𝐷 and 𝐶⨁𝐷 𝑀. But 𝐷 is a fully invariant submodule in 𝑀 implies, 𝐷 𝐷⋂𝑀 ⨁ 𝐷⋂𝑀 , such that 𝐷⋂𝑀 is fully invariant in 𝑀 and 𝐷⋂𝑀 𝑀 since 𝑀 𝐷. Hence 𝐷 𝐷⋂𝑀 ⨁𝑀 and 𝐷⋂𝑀 ⨁ 𝑀 . Now 𝐶⨁𝐷 𝐶⨁ 𝐷⋂𝑀 ⨁𝑀 𝑀 𝑀 ⨁𝑀 . Hence 𝐶⨁ 𝐷⋂𝑀 𝑀 . Thus 𝑀 satisfies condition (5) of Theorem (3.9), which implies 𝑀 is strongly type-𝑇 module. □ For 𝑅-modules 𝑁 and 𝐴. 𝑁 is said to be 𝐴- projective, if every submodule 𝑋 of 𝐴, any homomorphism ∅: 𝑁 ⟼ can be lifted to a homorphism, 𝜓: 𝑁 ⟼ 𝐴, that is if 𝜋: 𝐴 ⟼ , be the-natural epiomorphism, then there exists a homorphism 𝜓: 𝑁 ⟼ 𝐴 such that 𝜋 ∘ 𝜓 ∅. ∅𝜓 A  A/X  N    IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1808 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|361    𝑀 is called projective if 𝑀 is 𝑁-projective for every 𝑅-module 𝑁. If 𝑀 is 𝑀-projective, 𝑀 is called self-projective. [11] Proposition (3.15): If an 𝑅-module 𝑀 is strongly 𝑇 -type module and 𝐿 is fully invariant direct summand of 𝑀, then 𝐿 is strongly 𝑇 -type module and is strongly 𝑇 -type module if 𝑀 is self- projective. Proof: To prove 𝐿 is strongly 𝑇 -type module. Let 𝐴 be a submodule of 𝐿, hence 𝐴 is a submodule of 𝑀, and so by condition (3) of Theorem 3.9, there exists a fully invariant direct summand 𝐷 of 𝑀, such that 𝐴⨁𝐷 𝑀. Hence 𝐴⨁𝐷 ∩ 𝐿 𝐿 and so 𝐴⨁ 𝐷 ∩ 𝐿 𝐿. On the other hand, 𝐷 ⨁ 𝑀 implies 𝑀 𝐷⨁𝐷 for some 𝐷 𝑀. As 𝐿 is fully invariant submodule in 𝑀, 𝐿 𝐷 ∩ 𝐿 ⨁ 𝐷 ∩ 𝐿 , where 𝐷 ∩ 𝐿 is fully invariant in 𝐷, 𝐷 ⋂𝐿 is fully invariant submodule in 𝐷 .Now 𝐷⋂𝐿 is fully invariant in 𝐷 and 𝐷 is fully invariant in 𝑀, so 𝐷⋂𝐿 is fully invariant in 𝑀. Also 𝐷 ∩ 𝐿 ⨁ 𝐿, 𝐿 ⨁ 𝑀, so 𝐷 ∩ 𝐿 ⨁ 𝑀. Thus by [4, Lemma 2.3] 𝐷 ∩ 𝐿 is fully invariant direct summand of 𝑀. Let be a t-closed submodule in . Then 𝐶 is a t-closed in 𝑀. As 𝑀 is strongly 𝑇 -type module there exists a fully invariant direct summand 𝐷 of 𝑀 such that 𝐶⨁𝐷 𝑀 by Theorem 3.9. Let 𝑀 𝐷⨁𝐷 for some 𝐷 𝑀 and since 𝐿 is fully invariant in 𝑀, 𝐿 𝐷⋂𝐿 ⨁ 𝐷 ⋂𝐿 such that 𝐷⋂𝐿 is fully invariant in 𝐷, 𝐷 ⋂𝐿 is fully invariant in 𝐷 . Then ⨁ ⋂ ⨁ ⋂ ≃ ⋂ ⨁ ⋂ ≃ ⨁ . But it is easy to see that ⨁ . As 𝐿 ⨁ 𝑀, 𝐿 is closed and this implies that ⨁ by [8,Proposition 1.4,P.18]. Thus ⨁ . On the other hand, since 𝐷 is a fully invariant submodule in 𝑀 and, 𝐿 is fully invariant in 𝑀, then 𝐷⨁𝐿 is fully invariant in 𝑀. Hence is fully invariant in by [16, Lemma 1.1.20(2)] (since 𝑀 is self-projective). Thus is a fully invariant direct summand of and ⨁ . Therefore is strongly 𝑇 -type module by Theorem 3.9(13). □ Corollary (3.16): If 𝑅 is a commutative strongly 𝑇 -type module and 𝐿 ⨁ 𝑅, then is a strongly 𝑇 -type module. Corollary (3.17): Let 𝑀 be a multiplication strongly 𝑇 -type module and 𝐿 ⨁ 𝑀. Then is strongly 𝑇 -type modu References [1] M. S. Abas On Fully Stable Modules Ph.D Thesis College of Science University of Baghdad. 1991 [2] Sh. Asgari and A. Haghany t-Extending modules and t-Baer modules Comm.Algebra 39 1605-1623. 2011 [3] Sh. Asgari, A. Haghany and Y Tolooei . T-semisimple modules and T-semisimple rings comm Algebra 41 5 .1882-1902. 2013 [4] Sh. Asgari, A. 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