ON HEREDITARY NOETHERIAN AND PRIME MODULES OVER LEAVITT PATH ALGEBRA: SINK AND INFINITE EMITTER
DOI:
https://doi.org/10.25077/jmua.14.4.390-399.2025Keywords:
HNP module, Leavitt path algebra, SinkAbstract
This study explores the structural and homological characteristics of left modules over Leavitt path algebras, focusing on those generated by specific vertex types in a directed graph, particularly sinks and infinite emitters. The paper examines the module \( L_K(E)u \), where \( u \in E^0 \) represents either a sink or an infinite emitter, and determines whether this module exhibits the properties of being hereditary, Noetherian, and prime. Our findings indicate that \( L_K(E)u \) is indeed hereditary, implying that all its submodules are projective. Additionally, the module satisfies the ascending chain condition, making it Noetherian. However, it fails to qualify as an a-prime module, since there exist nontrivial left ideals \( I \subsetneq L_K(E) \) for which \( IM \subsetneq M \), thus does not meet the criteria for primality. These results emphasize how the presence of sinks and infinite emitters significantly affects the module-theoretic behavior of Leavitt path algebras.
Downloads
References
[1] G. Abrams, F. Mantese and A. Tonolo, Extensions of simple modules over Leavitt
path algebras, Journal of Algebra 431, (2015), 78-106.
[2] P. Ara, M.A. Moreno, E. Pardo, Non-stable K-theory for graph algebras, Algebr.
Represent. Theory 10,(2007), 157-178.
[3] P. Ara, M. Brustenga, Module theory over Leavitt path algebras and K-theory,
J. Pure Appl. Algebra 214, (2010), 1131-1151.
[4] P. Ara, K.M. Rangaswamy, Finitely presented simple modules over Leavitt path
algebras, J. Algebra 417, (2014), 333-352.
[5] P. Ara, K. Goodearl, Leavitt Path Algebras of Separated , J.f¨ur die reine und
angewandte Mathematik, (2011), 135-143.
[6] G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293, (2005),
319 - 334.
[7] G. Aranda Pino, D. Martin Barquero, C. Martin Gonzalez, M. Siles Molina,
Socle theory for Leavitt path algebras of arbitrary graphs, Rev. Mat. Iberoam,
26, (2010), 611-638.
[8] I.Assem, D.Simson, A.Skowro´nski, Elements of the Representation Theory of
Associative Algebras 1, London Mathematical Society Student Texts 65, Cambridge
University Press, (2006).
[9] X.W. Chen, Irreducible representations of Leavitt path algebras, Forum Math,
20, (2012).
[10] J. Dauns, Modules and Rings, Cambridge university press, (1994).
[11] Irawati, Struktur Modul di Sekitar Modul Dibangun secara Hingga atas
Gelanggang HNP dan Generalisasi Gelanggang HNP, Disertasi, Departemen
Matematika ITB, (2004).
12] Irawati, The generalization of HNP ring and finitely generated modules over
HNP ring, International Journal of Algebra vol 5 no 13, (2011), 611-626.
[13] N. Jacobson, Structure of Rings vol 37, Amer. Math. Soc, (1956).
[14] C Lomp and A. J. Pena, A Note on Prime Modules, Divulgaciones Matematicas
vol 8 (2000), 1,31-42.
[15] W.G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 42
(1962),113.
[16] B. Pareigis, Categories and Functors, Academic Press, (1970).
[17] K.M. Rangaswamy,The theory of prime ideals of Leavitt path algebras over
arbitrary graphs, J. Algebra vol. 375, (2013), 73 - 96.
[18] K.M. Rangaswamy, On simple modules over Leavitt path algebras, J. Algebra
vol. 423, (2015), 239 - 258.
[19] K.M. Rangaswamy, A Survey of Some of the Recent Developments in Leavitt
Path Algebras, (2018).
[20] K. Saleh, P.Astuti, I. Muchtadi-Alamsyah, On the structure of finitely generated
primary modules, JP Journal of Algebra, Number Theory and Applications
vol 38 Issue 5 (2016), 519-533.
[21] N. V. Sanh, An Introduction to some kinds of Radicals in Module Theory,
SouthEast Asian conference on Mathematics and its Applications, (2013).
[22] I.G.A.W.Wardhana, P.Astuti, I. Muchtadi-Alamsyah, On almost prime submodules
of a module over a principal ideal domain, JP Journal of Algebra,
Number Theory and Applications vol 38 Issue 2, (2016), 121-128.
[23] I. E. Wijayanti, Coprime Modules and Comodules. Journal of Algebra, vol 305,
number 2,(2006), 937-948.
[24] Risnawita, F. Savoji, D. D. Wulandari, Modul Herediter Noether dan Prima
(HNP) Dari Modul Sederhana Chen A. Lattice Journal, vol 4, (2024).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Departemen Matematika dan Sains Data FMIPA UNAND
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
All articles published in Jurnal Matematika UNAND (JMUA) are open access and licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). This license allows published content to be freely accessed, shared, redistributed, and adapted for any purpose, including commercial use, provided that the applicable license conditions are followed.
Copyright Holder
The copyright of articles published in Jurnal Matematika UNAND is held by the Departemen Matematika dan Sains Data, Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA), Universitas Andalas (UNAND), in accordance with the journal's stated copyright policy.
Author Rights
Under the CC BY-SA 4.0 license, authors may share, distribute, and reuse their published articles, subject to the applicable license conditions. These rights include:
-
Self-Archiving: Authors may deposit their published articles in institutional repositories, personal websites, or academic platforms.
-
Reuse: Authors may reuse their published work in subsequent publications, research, teaching materials, or other lawful purposes, provided that appropriate attribution is maintained.
-
Distribution: Authors may share and distribute their published articles without requiring additional permission under the terms of the license.
Creative Commons License (CC BY-SA 4.0)
Under the CC BY-SA 4.0 license, users are free to:
-
Share: Copy and redistribute the material in any medium or format.
-
Adapt: Remix, transform, and build upon the material for any purpose, including commercial purposes.
These permissions are subject to the following conditions:
-
Attribution: Users must give appropriate credit to the original author(s), retain relevant copyright notices, provide a link to the license, and indicate if changes were made. Attribution must not imply endorsement by the author(s), copyright holder, or journal.
-
ShareAlike: If users remix, transform, or build upon the material, they must distribute their contributions under the same CC BY-SA 4.0 license or a compatible license.
-
No Additional Restrictions: Users may not apply legal terms or technological measures that legally restrict others from exercising the rights granted by the license.
For more information about the CC BY-SA 4.0 license, please visit:
https://creativecommons.org/licenses/by-sa/4.0/
Third-Party Content
Third-party materials, including figures, tables, photographs, and illustrations, are covered by the article's CC BY-SA 4.0 license only when their inclusion under that license is authorized.
Materials excluded from the license must be clearly identified with appropriate copyright notices and attribution. Authors are responsible for obtaining any necessary permissions, while users must comply with the applicable licensing terms or permissions when reusing excluded materials.
Copyright and Licensing Information
Copyright and licensing information should be clearly displayed on each published article's landing page and included in the full-text versions (HTML and PDF), where applicable, to ensure transparency regarding copyright ownership and permitted reuse.
