On n-layered QT AG-modules

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1 Bull. ath. Sci : DOI /s x On n-layered QT AG-modules Fahad Sikander Ayazul Hasan Alveera ehdi Received: 24 December 2013 / Revised: 23 arch 2014 / Accepted: 3 April 2014 / Published online: 20 April 2014 The Authors This article is published with open access at SpringerLink.com Abstract A module over an associative ring with unity is a QT AG-module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. There are many fascinating concepts related to these modules. Here we introduce the notion of n-layered QT AG-modules and discuss some interesting properties of these modules. We show that a QT AG-module is n-layered if and only if / is an n-layered module, whenever is a finitely generated submodule of and n 1 is an integer. Keywords QTAG-module ω-elongation Totally projective modules ω + k-projective modules athematics Subject Classification K20 Communicated by S.K. Jain. F. Sikander B College of Computing and Informatics, Saudi Electronic University Jeddah Branch, Jeddah 23442, Kingdom of Saudi Arabia fahadsikander@gmail.com; f.sikander@seu.edu.sa A. Hasan Department of athematics, Integral University, Lucknow , India ayaz.maths@gmail.com A. ehdi Department of athematics, Aligarh uslim University, Aligarh , India alveera_mehdi@rediffmail.com

2 200 F. Sikander et al. 1 Introduction and preliminaries The study of QTAG-modules was initiated by Singh [8]. ehdi et al. [4] workeda lot on these modules. They studied different notions and structures of QTAG-modules and developed the theory of these modules by introducing several notions, investigated some interesting properties and characterized them. Yet there is much to explore. Throughout this paper, all rings are associative with unity and modules are unital QT AG-modules. An element x is uniform, if xr is a non-zero uniform hence uniserial module and for any R-module with a unique composition series, d denotes its composition length. For a uniform element x, ex = dxr and H x = sup{d yr xr y, x yr and y uniform} are the exponent and height of x in, respectively. H k denotes the submodule of generated by the elements of height at least k and H k is the submodule of generated by the elements of exponents at most k. A submodule of is h-pure in if H k = H k, for every integer k 0. A submodule of a QTAG-module is height finite, if the heights of the elements of take only finitely many values. is h-divisible if = 1 = k=0 H k and it is h-reduced if it does not contain any h-divisible submodule. In other words it is free from the elements of infinite height. A submodule is nice [3, Definition 2.3] in, if H σ / = H σ + / for all ordinals σ, i.e. every coset of modulo may be represented by an element of the same height. A family of nice submodules of submodules of is called a nice system in if i 0 ; ii If { i } i I is any subset of, then I i ; iii Given any and any countable subset X of, there exists K containing X, such that K/ is countably generated [4]. A h-reduced QT AG-module is called totally projective if it has a nice system. For a QT AG-module, there is a chain of submodules τ = 0, for some ordinal τ. σ +1 = σ 1, where σ is the σ th-ulm submodule of. A fully invariant submodule L is a large submodule of, if L + B = for every basic submodule B of. Several results which hold for TAG-modules also hold good for QT AG-modules [8]. otations and terminology are follows from [1,2]. 2 n-layered QT AG-modules and its properties Recall that a QT AG-module is ω + 1-projective if there exists submodule H 1 such that / is a direct sum of uniserial modules and a QT AG module is ω + k-projective if there exists a submodule H k such that / is a direct sum of uniserial modules [4]. Let σ be a limit ordinal such that σ = ω + β. A QT AG-module is called σ - projective, if there exists a submodule H β such that / is a direct sum of uniserial modules. A QTAG-module is totally projective, if and only if /H σ is σ -projective for every ordinal σ.

3 On n-layered QT AG-modules 201 A QT AG-module is an ω-elongation of a totally projective QT AG-module by a ω + k-projective QT AG-module if and only if H ω is totally projective and /H ω is ω + k-projective. A QT AG-module is a strong ω-elongation of a totally projective module by a ω+n-projective module if H ω is totally projective and there exists H n +H ω such that is a direct sum of uniserial modules [5]. Referring to our criterion from [7], is a -module or layered module if Soc = k<ω k, where k k+1 Soc and for every k, k H k = SocH ω. In [5], it was shown that any ω + 1-projective σ -module is a direct sum of countable modules of length atmost ω + 1. oreover, we extended this assertion to the so called strong ω-elongations. It was established that any strong ω-elongation of a totally projective module by a ω + 1-projective module is a -module precisely when it is totally projective. That is why it naturally comes under what additional conditions on the module structure this type of results hold for every n. To achieve this goal we state the following new concept, which is a generalization of the corresponding one for -module. Definition 1 A QT AG-module is said to be n-layered module if for some n <ω, H n = k<ω k, k k+1 H n and for all k 1, k H k = H n H ω. Remark 1 Equivalently, we may say that is a n-layered module if and only if H n = k, k k+1 H n and for every k 1, k H k H ω. Also, k k + H n H ω implies that H n = k + H n H ω and k + H n H ω H k = H n H ω + k H k = H n H ω. Therefore k = k + H n Hω and k H k H ω, equivalently k H k = H n H ω. Remark 2 Every layered module is 1-layered module and vice-versa. Since H n H m,forn m,everym-layered module is a n-layered module. ow we investigate some properties of n-layered modules. Lemma 1 For n 1, h-pure submodules of n-layered modules are n-layered modules. oreover, the submodules of n-layered modules with the same first Ulm submodules are n-layered. Proof Let be a n-layered QT AG-module such that H n = j<ω j, j j+1 H n and j H j H ω. owforanyh-pure submodule of, H n = j<ω j, where j = j and and the result follows. j H j H ω = H ω

4 202 F. Sikander et al. If K is an arbitrary submodule of such that H ω K = H ω, then H n K = j<ω K j, where K j = j K and and we are done. K j H j K K H ω = H ω K Lemma 2 Let be submodule of a h-reduced module and n 1. Then is n-layered if and only if is n-layered, where is a H ω+n 1 -high submodule of. Proof For any ordinal α, H α -high submodules of are h-pure in. owif is a H ω+n 1 -high submodule of, itish-pure in and by Lemma 1, if is n-layered, then is also n-layered. We have Let be a h-reduced QTAG-module and let be a H α+k -high submodule of with α a limit ordinal and k 1. Then H n = H n H n K for n >k and any complementary summand K of a maximal summand of H α bounded by k [4]. ow for the converse, we have H n = H n H n K, where K is a H n 1 H ω -high submodule of H ω. Since H n = j<ω j, j j+1 H n and j H j H ω, by defining j = j H n K, wehave H n = j<ω j. Since is h-pure in and K H ω, j H j = H n K + j H j = H n K + j H j H ω. Proposition 1 For n 1, all -modules with h-divisible first Ulm submodule are n-layered modules. Proof Let be a -module such that H ω is h-divisible. Since h-divisible submodules are direct summands, we have = H ω, where is contained in a high submodule of, hence is a direct sum of uniserial submodules. Again H ω and we are done. Proposition 2 Direct sums of n-layered modules are n-layered modules. Proof Let be a direct sum of n-layered modules such that = j J j.here j s are n-layered modules. Therefore H n j = i<ω ij, ij i+1 j j and ij H i j H ω j for i <ω,j J. Furthermore, H n = j J H n j = j J i<ω ij= i<ω j J ij= i<ω i, where i = j J ij and i H i = j I ij H i j j I = ij H i j j I j I H ω j

5 On n-layered QT AG-modules 203 = H ω j I j = H ω and the result follows. Proposition 3 For k 1, is a n-layered module if and only if H k is n-layered. Proof If is a n-layered module, then H n = j<ω j, j j+1 H n and j H j H ω, for every j <ω. Therefore H n H k = i<ω T j, where T j = j H k and T j H k+ j = T j H j H k j H j H ω = H ω H k. Thus H k is also n-layered. For the converse, suppose H k is n-layered. If k=1, then H 1 is n-layered and we have H n H 1 = j<ω T j, T j T j+1 H n H 1 and T j H j+1 H ω.lets ={x x H n, x / H n H 1 }. Then H n = S H n H 1. Define K j such that K j H 1 = φ and H n \H 1 = j<ω K j such that K j H 1 T j. This implies that H n = T j + K j, and T j + K j H j+1 T j + K j H 1 = T j + K j H 1 = T j. Therefore T j + K j H j+1 T j H j+1 H ω and the result follows. Proposition 4 A QT AG-module is n-layered module if and only if its large submodule L is n-layered. Proof For a large submodule L of, H ω L = H ω [6]. Therefore by Lemma 1, L is n-layered whenever is n-layered. Conversely, suppose L is n-layered such that L = k<ω H k H mk, where m 1 m 2 m k is a monotonically increasing sequence of positive integers. ow H n L = SocH m1 + + H n H mn therefore H n H mn H n L H n H m1. Also H n L = j<ω L j, L j L j+1 H n L and L j H j L H ω L and H n H mn = j<ω j, where j = L j H n H mn = L j H mn. Again j H t j j H j L H ω L = H ω,forsomet j max j, m n with H n H t j H n H j L, ash j L is also a large submodule of. ow H mn is n-layered module and by Proposition 3, is also n-layered module.

6 204 F. Sikander et al. Proposition 5 Let be a submodule of such that / is bounded. Then is n-layered module if and only if is n-layered module. Proof Since / is bounded, then there exists an integer k such that H k / = 0 or H k. Therefore H ω H k = H ω = H ω and by Lemma 1,if is n-layered then is also n-layered. Conversely, if is a n-layered module then by Lemma 1, H k is also n-layered. Therefore by Proposition 3, is also n-layered. Proposition 6 Let be a height-finite, submodule of. If / is n-layered, then is n-layered. Proof Since / is n-layered, H n / = j<ω K j/ = K j /, where K j K j+1 and H j Hω.ow is height-finite, therefore K j nice in and K j H j H ω +. There exists a positive integer t j j such that H t j H ω.also K j H t j H ω + H t j = H ω + H t j = H ω. ow H n + H n / and H n j<ω K j. Thus H n = j<ω T j, where T j = K j H n = H n K j and the result follows. Remark 3 Let be a height-finite submodule of.if/ is a -module, then is also a -module. Proposition 7 Let be a submodule of. i if H n = H n and is finitely generated or H ω and is n-layered, then / is also n-layered; ii if H k,forsomek 1 and either is finitely generated or H ω and is n + k-layered, then / is also n-layered. Proof i If is n-layered, then H n = j<ω j, j j+1 and j H j H ω. ow,h n = H n + = j + j<ω. Therefore, j + H [+ j = j + H j ]. When H j, for every positive integer j, then j + H j + j H j + H ω. Since +Hω H ω, the result follows. When is finitely generated, then there exists an integer t j j such that j + H t j H ω. Therefore j + H t j +Hω = H ω and we are done.

7 On n-layered QT AG-modules 205 If H ω and we have H n = j j<ω, j j+1, where j / H j / H ω / = H ω /. Therefore j H j H n + H ω. Since follows. ii Since H n H n+k + H n, H n = j<ω H n j and the result, we are through. Proposition 8 If for some ordinal α, /H α is n-layered, then is n-layered. Proof We have H n /H α = j<ω j/h α, j j+1, j H j H ω for every j <ω.ow H n + H α H n H α H α therefore H n j<ω j. If we put T j = H n j, then H n = j<ω T j. But T j H j j H j H ω and we are done. ow we are in the state to prove our main result which motivated this article. Theorem 1 The QT AG-module is a n-layered module which is a strong ω-elongation of a totally projective module by a ω + n-projective module if and only if is a totally projective module. Proof Since is a strong ω-elongation, H ω is totally projective and there exists a submodule H n such that +H ω is a direct sum of uniserial modules /H and +H ω ω +H ω /H ω. ow by the definition of n-layered modules, H n = j<ω j, j j+1 H n and j H j = H n H ω,for every j. Since H n, = j<ω j where j = j and +Hω H ω =. j<ω j +H ω H ω ow, j + H ω H ω H j H ω = j + H ω H ω H j, H ω = [ j + H ω H j ], H ω = H ω + j H j, H ω = 0. Therefore /H ω is a direct sum of uniserial modules and is totally projective. We have shown that if is a finite submodule of such that H n = H n, then is an n-layered module if and only if / is an n-layered module. oreover, in [7] we showed that is -module if and only if / is a -module. We generalize this assertion to n-layered modules for an arbitrary natural number n. For doing this, we need following technical lemmas:

8 206 F. Sikander et al. Lemma 3 Let be a finitely generated submodule of. Then for an integer n 1, H n / = H n + K where K is a finitely generated submodule of with H n K K. Proof Let x + H k / for some x such that there exists y with d xr yr = n. We may express H n = m i=1 x i R for some m Z + and put K = + y i R, where d yi R x i R = n. If y k such that k = 1,...,m and x k such that d yk R x k R = n, then x k R = x i R for some i {1, 2,...,m}. Therefore y k R y i R + H n K + H n. The converse is trivial and the result follows. Lemma 4 Let K be a h-finite submodule of having only finite heights in. If is a finitely generated submodule of then + K is also h-finite assuming finite heights only. Proof Since the elements of K assumes only finite number of finite heights, K H k 1, for some k 1. ow is finitely generated submodule and we may express as m i=1 x i R. Consider the submodule of where = t i=1 x i R such that x i + y i H ni but x i + y i / H ni+1 with n i > k, i = 1, 2,...,t for some y i K. Therefore for each y K we have y + x i = y y i + y i + x i / H ni +1, otherwise y y i + y i + x i H ni implying that y y i H k and y y i 1. Therefore y i + x i H ni +1 which is a contradiction whenever 1 i t. If we put n = max{n 1 + 1,...,n t + 1}, y + x i / H n. Since y + x j / H n for t + 1 j n, we are done. ow we are ready to prove our main result: Theorem 2 For each natural number n, a QT AG-module is n-layered if and only if / is n-layered, where is a finitely generated submodule of. Proof Suppose that is an n-layered QTAG-module, then H n = i<ω i, i i+1 H n and, for all i <ω i H i 1. By Lemma 3 we may write H n / = H n + K /, for some finitely generated submodule K of containing. Furthermore, H n / = i<ω i + K / and by Lemma 4, we calculate that

9 On n-layered QT AG-modules 207 i + K Hti i + K = Hti + = i + K H ti + = i + K H ti for every i and some natural number t i i, implying that / is an n-layered module. For reverse implication, suppose that / is an n-layered module. ow write H n / = Ti i<ω, T i T i+1 and for all i <ω, Ti Hi = 1. Since is finitely generated it is nice in. ow we may say H n + H n, Ti i<ω i<ω = T i and =. Therefore H n = i<ω H n T i and Ti Hi + = 1 +. Therefore T i H i + = T i H i = 1 + T i H i 1 +. and Since is finitely generated so there exists m such that H m 1, therefore T i H ti 1 + H m 1 + H m = 1, for every i and t i = m + i, implying that is also n-layered. Acknowledgments The authors are thankful to the referee for his/her valuable suggestions. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original authors and the source are credited.

10 208 F. Sikander et al. References 1. Fuchs, L.: Infinite Abelian Groups, vol. I. Academic Press, ew York Fuchs, L.: Infinite Abelian Groups, vol. II. Academic Press, ew York ehdi, A., Abbasi,.Y., ehdi, F.: ice decomposition series and rich modules. South East Asian J. ath. ath. Sci. 41, ehdi, A., Abbasi,.Y., ehdi, F.: On ω + n-projective modules. Ganita Sandesh 201, ehdi, A., Skander, F., aji Sabah, A.R.K.: On elongations of QTAG-modules. ath. Sci. accepted for publication. doi: / / ehran Hefzi, A., Singh, S.: Ulm-Kaplansky invariants of TAG-modules. Commun. Algebra 132, aji, S.A.R.K.: A study of different structures of QT AG-modules. Ph.D. Thesis, A..U., Aligarh Singh, S.: Some decomposition theorems in abelian groups and their generalizations. In: Ring Theory: Proceedings of Ohio University Conference, vol. 25, pp arcel Dekker, Y 1976