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The Scientific World Journal
Volume 2013 (2013), Article ID 384131, 3 pages
A Note on Elongations of Summable QTAG-Modules
1Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
2Department of Mathematics, Al-Bayda University, Al-Bayda, Yemen
Received 15 August 2013; Accepted 14 October 2013
Academic Editors: S. Khan and A. Mimouni
Copyright © 2013 Alveera Mehdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A right module over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. In this paper we find a suitable condition under which a special -elongation of a summable QTAG-module by a -projective QTAG-module is also a summable QTAG-module.
1. Introduction and Preliminaries
The study of QTAG-modules was initiated by Singh . Mehdi et al. studied different notions and structures on QTAG-modules, developed the theory of these modules by introducing several notions and some interesting properties of these modules and characterized different submodules of QTAG-modules. Yet there is much to explore.
Throughout this paper, all rings will be associative with unity and modules are unital QTAG-modules. An element is uniform, if is a nonzero uniform (hence uniserial) module and for any -module with a unique composition series, denotes its composition length. For a uniform element , and are the exponent and height of in , respectively. denotes the submodule of generated by the elements of height at least and is the submodule of generated by the elements of exponents at most . A QTAG-module is called a -module, if all of its high submodules are direct sums of uniserial modules. is -divisible if .
A module is summable if , where is the set of all elements of which are not in , where is the length of .
A submodule is nice [2, Definition 2.3] in , if for all ordinals ; that is, every coset of modulo may be represented by an element of the same height.
Recall that a QTAG-module is -projective if there exists a submodule such that is a direct sum of uniserial modules and a QTAG-module is -projective if there exists a submodule such that is a direct sum of uniserial modules .
A QTAG-module is an -elongation of a totally projective QTAG-module by a -projective QTAG-module if and only if is totally projective and is -projective.
2. Main Results
In , we studied strong -elongations of summable modules by -projective QTAG-modules. It was found that the module is a special -elongation if is summable and there exists a submodule such that is a direct sum of uniserial modules. We showed there that under certain additional circumstances on these elongations are of necessity summable modules; in fact was taken; that is, is a height finite submodule of .
In , the class of -layered module was introduced which is a proper subclass of -modules as follows: is an -layered module if , , and for all . . It was proved there that every -layered module which is a strong -elongation of a totally projective module by a -projective module is totally projective and vice versa; in particular each -layered module is -projective uniquely when it is a direct sum of modules of length at most .
Motivated by these elongations we investigate the relationship between the classes of -layered modules and strong -elongations of summable modules by -projective modules, that is, how -layered modules are situated inside these special -elongations of summable modules by -projective modules and whether there is an analogue with the strong -elongations of a totally projective modules by -projective modules.
Before doing that, we need the following preliminaries.
A module is said to be pillared , if is a direct sum of uniserial modules. Clearly, such a module is necessarily -layered module and hence a -module , whereas the converse is not true in general. We investigate the conditions under which the converse holds good.
A module is said to be a strong -elongation (of a summable module) by a -projective module if there exists a submodule with a direct sum of uniserial modules. The first Ulm-factor of this module is -projective. There are modules with -projective first Ulm-factor which are not strong -elongations by a -projective module. These modules are with strongly -projective first Ulm-factor.
Now we are in the state to prove the following main result.
Theorem 1. An -layered module is a strong -elongation by a -projective module if and only if it is a pillared module.
Proof. Suppose that is an -layered module, so , , and for all . and suppose that there exists a submodule of such that with is a direct sum of uniserial modules. Clearly,
Since , we infer that and thus by putting .
Using modular law, we compute that Hence is a direct sum of uniserial modules and therefore is pillared. The proof is complete as the converse is trivial.
Proposition 2. An -layered module is a strong -elongation of a summable module by a -projective module if and only if it is a summable pillared module.
Proof. Suppose -layered module is a strong -elongation of a summable module by a -projective module. Since is a -module and is summable, has to be summable as well and Theorem 1 ensures that must be pillared.
Suppose that is summable pillared module, so it ensures that is summable and pillared modules are both -layered and strong -elongations by -projective module by taking , which completes the proof.
As an immediate consequence of the above, we have the following corollary.
Corollary 3. Suppose is a -module which is a strong -elongation of a summable module by a -projective module and the th Ulm-Kaplansky invariants of are zero for each such that , if . Then is a summable pillared module.
Proof. Since th Ulm-Kaplansky invariants of are zero, where is a high submodule of . Since it is a direct sum of uniserial modules, where . Putting , we obtain . Since is -pure in , we have By Proposition 2, is an -layered module.
Proposition 4. A module of length not exceeding is an -layered module if and only if it is a direct sum of countably generated modules.
Proof. Suppose is an -layered module of length not exceeding ; then and hence
since . Moreover, we write , and . Consequently, , where . We compute
hence is pillared and as is bounded, is a direct sum of countable modules.
The sufficiency is obvious and follows from .
As an immediate consequence of the above, we have the following corollary.
Corollary 5. A module is an -layered module if and only if each of its -high submodules is a direct sum of countable modules.
Proof. Let be an -layered module and be its -high submodule. In , we show that is an -layered module precisely when is an -layered module. Hence using the Proposition 4, result follows immediately.
With the help of the last statement we can verify once again the validity of Corollary 3 as it is checked that a submodule of is -high in if and only if is -high in whenever the th Ulm-Kaplansky invariants of are zero for ; that is,
A module is said to be strong -elongation of a summable module by a totally projective module if is summable and there is a nice submodule of such that and is totally projective .
Now we are in the state to prove our final result.
Theorem 6. An -layered module is a strong -elongation of a summable module by a totally projective module if and only if it is a summable pillared module.
is totally projective. Since , is contained in some -high submodule of , say . By Corollary 5, is totally projective of length at most , so we may write , where and all ’s are height finite in and hence in as is isotype in . Therefore
In the same way, we can show that are height finite in and by , is nice in and hence is totally projective. Then
should be a direct sum of uniserial modules and hence is pillared.
On the other hand, being summable implies that is summable and hence has to be summable; thus it is summable pillared, which completes the proof.
As an immediate consequence for , we have the following.
Corollary 7. A -module is a strong -elongation of a summable module by a totally projective module if and only if it is a summable pillared module.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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