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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 10, Issue 3 Ver. I (May-Jun. 2014), PP 53-61 www.iosrjournals.org Topological Projective Modules of topological ring R Dr. Taghreed Hur Majeed Department of Mathematics- College of Education – Al-Mustansiriya University Abstract: Our principle aim to study an algebraic module properties topolocically. We concern topological projective modules specially. A topological module P is called topological projective module if for all topological module epimorphism g : B A and for all topological module morphism f :P B, there exist a topological module morphism f*: P A, for which the following diagram commutes: P f A g B 0 The important point of this paper is to find topological projective module from topological module given. Furthermore among results are obtained at the end of the paper. Key words: Topological module, topological ring, topological submodule, topological projective module. I. Introduction Kaplanski is the first scientist who introduced the definition of topological module and topological submodule in 1955, after that a topological module studied by many scientists likes Alberto Tolono and Nelson. In this research a necessary and sufficient condition for topological module to be topological projective module has been established, the kind of topological don't determinant. The result in this paper algebraically satisfied, now we will satisfy it algebraically and topologically. This work divided into two sections. In section one includes some necessary definitions while section two includes propositions. We concern of this research is to obtain topological projective module from topological module given we suppose n, n1, n2, …, nm be natural number. 1- Topological Modules In this section we give the fundamental concepts of this work. Definition (1.1) : [3] A non empty set E is said to be topological group if: (1) E is a group. (2) is A Topology on E. (3) A mapping : E E E and : E E are continuous, where defined as (x) = x – 1 and E E is the product of two topological spaces. Definition (1.2) : [4] Let E be topological group. A subset M of E is said to be a topological subgroup of E if: (1) M is a subgroup of E. (2) M is a subspace of E. Definition (1.3) : [1] A non empty set R is said to be topological ring if: (1) R is a ring. (2) is A Topology on R. (3) A mapping : R R R defined as (x,y) = xy and : R R defined as (x) = x continuous. www.iosrjournals.org – 1 are 53 | Page Topological Projective Modules of topological ring R Definition (1.4): [1] Let R be a topological ring. A non empty set E is said to be topological module on R, if: 1. E is module on R. 2. E is a topological group. 3. A mapping :RE E defined as (,x) =x is continuous for all R, x E. Example (1.5): [2] Let E be a topological module on topological ring R, a subset M of E is said to be topological submodule of E if: 1. M is a submodule of E. 2. M is a topological subgroup of E. Proposition (1.6): [2] Let R be topological ring and {E}1 n be finite discrete topological module of R then the direct sum Eα 1 α n be discrete topological module. Corollary (1.7): [2] Let R be a topological ring, {E}∆ be a family of discrete topological module then the direct sum Eα α be discrete topological module. Definition (1.8): [3] Let E and E be two topological modules, the map f from E into E' is said to be homomorphism topological module if: 1. f is homomorphism module. 2. f is continuous map. Definition (1.9): [3] Let E and E be two topological modules on the map f from E into E is said to be homomorphism topological module, if: 1. f is homomorphism module. 2. f is topological homeomorphism. 2- Topological Free Modules Definition (2.1): [5] A topological module P is said to be topological projective module if for all topological module epimorphism g: B A and for all topological module morphism f: P B, there exists a topological module morphism f *: P A, for which the following diagram commutes: P f A g B Notation (2.2): [1] Ker f is topological submodule of E, where f is topological module homomorphism from E (topological module) into E' (topological module). Proposition (2.3): [3] Let P be a topological projective on a ring R and P be discrete topological module, then P is topological projective module. www.iosrjournals.org 54 | Page Topological Projective Modules of topological ring R Proposition (2.4): [3] If {Eα }α be a family of topological modules on topological ring R, B be a topological module on the same ring, the topological module homomorphism g: P B for all L, there exists a unique topological module homomorphism g: B Pα α for which the following diagram commutes: P g Pα g α B Proposition (2.5): [3] If {Pα }α be a family of topological modules of topological ring R, then the direct sum P = topological projective module if and only if if {Pα }α {Pα } be α is a topological projective module. Proposition (2.6): If P1, P2 and P3 be topological module of topological ring R such that f 1: P2 P1 and f 2: P3 P2 be two topological module homeomorphism, then there exist f 1∘ f 2: P3 P1 is topological module homeomorphism. Proof: 2: Since f 1: P2 P1 and f 2: P3 P2 be two topological module homeomorphism, then there exist f 1∘ f P3 P1 ( f 1∘ f 2)(p3) = f 1( f 2(p3)) , for all p3 P3 = f 1(p2) = p1 , for all p2 P2 , for all p1 P1 f 1∘ f 2 is continuous map because f 1 and f 2 are continuous maps. And ( f 1∘ f 2) – 1 is continuous map since f 11 and f 21 are continuous maps. Thus f 1∘ f 2 is topological module homeomorphism. f2 f1 f 1∘ f 2 Proposition (2.7): If P1, P2, P3, …, Pn be topological module of topological ring R such that f k: Pk + 1 Pk, 1 k n – 1 topological module homeomorphism, then there exist i: Pn P1 is topological module homeomorphism. Proof: We prove it on the same way of proposition (2.6). www.iosrjournals.org 55 | Page Topological Projective Modules of topological ring R 𝜄 h f g B Proposition (2.8): If P1, P2 and P3 be topological module of topological ring R such that P1 and P2 are topological projective modules and i : P3 P1 be topological module homeomorphism, then P3 is topological projective module too. Proof: We take the following diagram to obtain A and B are topological modules of R. f : P1 B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Since P1 is topological projective module, there exists f *: P1 A is topological module homeomorphism. Now we define h: P3 B by form h(p3) = f *∘ i(p3) , for all p3 P3 and g∘ h(p3) = g∘ f *∘ i(p3) = f ∘ i(p3) = f (p3). Thus P3 is topological projective module. Proposition (2.9): If P1, P2, P3, …, Pn be topological module of topological ring R such that P1, P2, P3, …, Pn – 1 be topological projective modules and i: Pn P1 be topological module homeomorphism, then Pn is topological projective module too. Proof: We take the following diagram to obtain A and B are topological modules of R. f : P1 B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Since P1 is topological projective module, there exists f *: P1 A is topological module homeomorphism. Now we define h: P3 B by h(p3) = f *∘ i(pn) , for all pn Pn where f1 ∘ f2 ∘…∘ fn – 2 ∘ fn – 1 = i and g∘ h(p3) = g∘ f *∘ i(pn) = f ∘ i(pn) = f (pn), for all pn Pn Thus Pn is topological projective module too. www.iosrjournals.org 56 | Page Topological Projective Modules of topological ring R Proposition (2.10): If P and Pα 1α m be two topological modules of topological ring R such that P be topological projective Pα module and i: 1α m P be topological module homeomorphism, then Pα is topological Pα be topological 1α m projective module too. Proof: We take the following diagram to obtain A and B are topological modules of R. f : P B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Since P is topological projective module, there exists f *: P A is topological module homeomorphism. Now we define h: Pα 1α m A by h(q) = f *∘ i(q) , for all q Pα , 1α m q = p1 p2 … pm and g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Pα Thus 1α m is topological projective module too. Proposition (2.11): If Pα 1α m and P be two topological modules of topological ring R such that projective module and i: P Pα 1α m 1α m be topological module homeomorphism, then P is topological projective module too. Proof: We look at the following diagram to obtain A and B are topological modules of R. f: Pα 1α m B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Since f *: Pα 1α m Pα 1α m is topological projective module, there exists A is topological module homeomorphism. Now we define h: P A by h(p) = f *∘ i(p) , for all p P and g∘ h(p) = g∘ f *∘ i(p) = f ∘ i(p) = f (p) , for all p P. Thus P is topological projective module too. Proposition (2.12): If Pα1 1α1 n1 and Pα2 1α 2 n 2 topological projective module and i: Pα2 1α 2 n 2 be two topological modules of topological ring R such that Pα2 1α 2 n 2 Pα1 1α1 n1 Pα1 1α1 n1 be be topological module homeomorphism, then is topological projective module too. Proof: We look at the following diagram to obtain A and B are topological modules of R. www.iosrjournals.org 57 | Page Topological Projective Modules of topological ring R f: Pα1 1α1 n1 B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Pα1 Since f *: 1α1 n1 Pα1 1α1 n1 is topological projective module, there exists A is topological module homeomorphism. Now we define h: Pα2 1α 2 n 2 A by Pα2 , h(q) = f *∘ i(q) , for all q 1α 2 n 2 q p1 p2 ... pn 2 where and g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Pα2 Thus is topological projective module too. 1α 2 n 2 Proposition (2.13): Pα1 , If Pα2 1α1 n1 Pα1 1α1 n1 1α 2 n 2 Pα2 and 1α 2 n 2 and Pα3 be topological modules of topological ring R such that 1α3 n 3 are topological projective modules and topological module homeomorphism, then Pα3 1α3 n 3 i: Pα3 1α3 n 3 Pα1 1α1 n1 be is topological projective module too. Proof: We look at the following diagram to obtain A and B are topological modules of R. f: Pα1 1α1 n1 B is topological module homeomorphism. And g: A B is surjective topological module homeomorphism. Pα1 Since f *: 1α1 n1 Pα1 1α1 n1 is topological projective module, there exists A is topological module homeomorphism. Now we define h: Pα3 1α3 n 3 h(q) = f *∘ i(q) , for all q where A by Pα3 , 1α3 n 3 q p1 p2 ... pn3 and g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Thus Pα3 1α3 n 3 is topological projective module. Proposition (2.14): If Pα1 , 1α1 n1 such that Pα2 , Pα3 , …, 1α 2 n 2 1α3 n 3 Pα1 , Pα2 , Pα3 , …, 1α1 n1 1α 2 n 2 1α3 n 3 Pαm 1α m n m be topological modules of topological ring R Pαm1 are topological projective modules and i: www.iosrjournals.org 58 | Page 1α m 1 n m 1 Topological Projective Modules of topological ring R Pαm 1α m n m Pα1 be topological module homeomorphism, then 1α1 n1 Pαm 1α m n m is topological projective module too. Proof: We look at the following diagram to obtain A and B are topological modules of R. f: Pα1 1α1 n1 B is topological module homeomorphism. And g: A B is surjective topological Since f *: Pα1 1α1 n1 Pα1 1α1 n1 is topological projective module, there exists A is topological module homeomorphism. Now we define h: Pαm 1α m n m h(q) = f *∘ i(q) , for all q A by Pαm , 1α m n m q p1 p2 ... pnm where and g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Thus g∘ h = f and Pαm 1α m n m Proposition (2.15): {Pα1 } If α1 and is topological projective module. {Pα2 } , α 2 be a families of topological modules of topological ring R such that {Pα1 } be topological projective module and α1 topological module homeomorphism, then i: {Pα2 } α 2 {Pα1 } be α1 {Pα2 } is topological projective module too. α 2 Proof: We look at the following diagram to obtain A and B are topological modules of R. f: {Pα1 } B is topological module homeomorphism. α1 And g: A B is surjective topological module homeomorphism. Since f *: {Pα1 } is topological projective module, there exists α1 {Pα1 } A is topological module homeomorphism. α1 Now we define h: {Pα2 } A by α 2 h(q) = f *∘ i(q) , for all q where {Pα2 } , α 2 q p1 p2 ... pnm and g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). www.iosrjournals.org 59 | Page Topological Projective Modules of topological ring R Thus g∘ h = f and {Pα2 } is topological projective module. α 2 Proposition (2.16): If {Pα1 }, {Pα2 } α1 such that α 2 and {Pα3 } , be a families of topological modules of topological ring R α3 {Pα1 } and {Pα2 } are topological projective modules and i: {Pα3 } {Pα1 } be α1 α3 α 2 topological module homeomorphism, then α1 {Pα3 } is topological projective module too. α3 Proof: We look at the following diagram to obtain A and B are topological modules of R. f: {Pα1 } B is topological module homeomorphism. α1 And g: A B is surjective topological module homeomorphism. Since {Pα1 } is topological projective module, α1 there exists f *: {Pα1 } A α1 is topological module homeomorphism. Now we define h: {Pα3 } A by α3 h(q) = f *∘ i(q) , for all q {Pα3 } and α3 g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Thus g∘ h = f and {Pα3 } is topological projective module. α3 Proposition (2.17): {Pα1 } , {Pα2 } , {Pα3 } , If α1 α3 α 2 topological ring R such that projective modules and i: …, {Pαm } α m be a families of topological modules of {Pα1 } , {Pα2 } , {Pα3 } , α1 {Pαm } α m α 2 α3 {Pα1 } α1 …, {Pαm 1 } αm 1 are topological be topological module homeomorphism, then {Pαm } is topological projective module too. α m Proof: We look at the following diagram to obtain A and B are topological modules of R. f: {Pα1 } B is topological module homeomorphism. α1 And g: A B is surjective topological module homeomorphism. Since f *: {Pα1 } is topological projective module, there exists α1 {Pα1 } A is topological module homeomorphism. α1 Now we define h: {Pαm } A by α m www.iosrjournals.org 60 | Page Topological Projective Modules of topological ring R h(q) = f *∘ i(q) , for all q {Pαm } and α m g∘ h(q) = g∘ f *∘ i(q) = f ∘ i(q) = f (q). Thus g∘ h = f and {Pαm } is topological projective module. α m References Al-Anbaki, H., 2005, “Topological Projective Modules”, M.SC. Thesis, College of Education, Al-Mustanseriyah University. Boschi, E., 2000, “Essential Subgroups of Topological Groups”, Comm. In algebra, 28(10), pp.4941-4970. Mahnoub, M.; 2002, “On Topological Modules, M.Sc. Thesis, College of Science, Al-Mustanseriyah University. Majeed, T.H., 2013, "On Some Result of Topological Projective Modules", DJPS, Vol.(9), No.(3). Majeed, T.H., 2014, "On On Injective Topological Modules", Journal of the College of Basic Education, Al-Mustansiriya University, Vol.(20), No.(82). [6]. Sahleh, H., 2006, “The v-Trace of Abelian Topological Groups”, A generalization, Guilan University, Int. J., Vol.1, No.(8), pp.381-388. [1]. [2]. [3]. [4]. [5]. www.iosrjournals.org 61 | Page