On essentially finitely indecomposable $$ QTAG $$ Q T A G -modules

Hasan, Ayazul

doi:10.1007/s13370-015-0318-7

Cited by 4 publications

(3 citation statements)

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“…We start by defining the following: In [3,6,9], respectively, HT -modules were defined with the aid of small homomorphisms and large submodules. Here we study these modules in the light of quasi-large submodules.…”

Section: Quasi-large Submodulesmentioning

confidence: 99%

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On quasi-large submodules of QTAG-modules

Hasan

2016

Asian-European J. Math.

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We generalize the notion of large submodules by introducing what we call a quasi-large submodule. Here, we examine two notions closely related to quasi-large submodules; namely, the notions of HT -modules and that of essentially finitely indecomposable modules. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of closed modules. Now, we are in a position to proceed by proving the following: Theorem 3.2. Let M be a QTAG-module. Then the following conditions are equivalent.be a direct sum of uniserial modules, and let T be a hpure submodule of M supported by Soc(N ). Then M/T is a direct sum of uniserial modules and M = T ⊕ Q where Q is a direct sum of uniserial modules. Now, since M is essentially finitely indecomposable there exists k ∈ Z + such that H k (Q) = 0. Therefore, Soc(H k (M )) ⊂ Soc(T ) = Soc(N ) ⊂ N . (iii) ⇒ (ii). In view of Theorem 3.1 (iii), we need only show that if M/N is a direct sum of uniserial modules, then H 1 (N ) contains a submodule K of M such that M/K is a direct sum of uniserial modules. Let M/N be a direct sum of uniserial modules, then N ⊃ Soc(H k (M )), thus by Theorem 2.3, Soc(N ) supports a h-pure submodule T of M . Let K = H 1 (T ∩ N ); then by Lemma 3.1, K ⊂ H 1 (N ) and M/K is a direct sum of uniserial modules. Regarding the above theorem, the following immediately follows: Corollary 3.1. Let M be a h-pure-complete QTAG-module. Then, M is essentially finitely indecomposable if and only if M is a HT -module. Corollary 3.2. Let M be a QTAG-module such that M is closed. Then M is a HT -module. Now we present here some interesting properties of closed modules and then use them to derive a result due to Mehdi et al. [9].Theorem 3.3. Let N be a closed bounded submodule of a separable QTAG-module M. Then M is closed if and only if M/N is closed. Proof. Suppose M is closed and assume N ⊂ Soc(M ). Then N supports a h-pure submodule K of M . Since N is closed, then K is closed and therefore M = K ⊕ T for some submodule T of M . Now M/N ∼ = (K/N ) ⊕ T , where T is closed and since K/N ∼ = H 1 (K) it is also closed; therefore M/N is closed. By induction we assume that the result is true for every closed N ⊂ Soc k (M ) for every closed module M . Let N be a closed submodule of M contained in Soc k+1 (M ). Then N/Soc(N ) is 1650033-6 Asian-European J. Math. 2016.09. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 06/15/16. For personal use only. On quasi-large submodules of QTAG-modules 1650033-7 Asian-European J. Math. 2016.09. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 06/15/16. For personal use only. A. Hasan

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Section: Quasi-large Submodulesmentioning

confidence: 99%

“…In [3] it was seen that essentially finitely indecomposable modules and HTmodules are equivalent properties. We prove here a weaker form of this conjecture.…”

Section: Definition 32mentioning

confidence: 99%

On quasi-large submodules of QTAG-modules

Hasan

2016

Asian-European J. Math.

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“…For a submodule N ⊆ M , the valuation of N induced by height in M is defined by v(x) = H M (x), the height of x in M , for all x ∈ N and N = K ⊕L is a valuated direct sum if v(k + ) = min{v(k), v( )} for all k ∈ K and ∈ L [2].…”

Section: Introduction Notation and Other Conventionsmentioning

confidence: 99%