Modul Herediter Noether dan Prima (HNP) Dari Modul Sederhana Chen A_`
DOI:
https://doi.org/10.30983/lattice.v4i1.8679Keywords:
Herediter, Noether, PrimaAbstract
Let K be a field and is line graph with infinite path. Let L is a Leavitt path algebra with correspondence with graph E. Suppose is a module over ring (written- module , a module is considered to be hereditary if all its submodules are projective. A module is called a Noetherian module if is a finitely generated module. Suppose M is a left module over the gelanggang (written - module ). A proper submodule of is said to be prime if with and implies or . In this paper we will look at the characteristics of the hereditary nooetherian and prime modules in the context of Leavitt path algebra. Let L is a Leavitt path algebra, where E is a line graphs with infinite path and M is a module over Leavitt path algebra L, we find that then M is simple modules which is hereditary noetherian and not prime modules.
Misalkan adalah suatu lapangan dan merupakan graf garis yang lintasannya tak terhingga. Misalkan adalah aljabar lintasan Leavitt yang berkorespondensi dengan graf Misalkan adalah modul di atas gelanggang (ditulis -modul ), modul dikatakan herediter jika semua submodulnya bersifat proyektif. Modul M disebut modul Noetherian jika adalah modul yang dibangun secara hingga. Misalkan adalah modul kiri atas gelanggang (ditulis -modul ). Suatu submodul sejati dari dikatakan prima jika dengan dan mengakibatkan atau . Dalam tulisan ini kita akan melihat karakteristik modul herediter noether dan prima dalam konteks aljabar lintasan Leavitt. Misalkan adalah aljabar lintasan Leavitt, dengan adalah graf garis yang memuat lintasan tak berhingga. Dan adalah modul atas aljabar lintasan Leavitt kita temukan bahwa M adalah modul sederhana yang merupakan modul herediter Noether dan bukan modul prima.
References
G. Aranda Pino, E. Pardo, and M. Siles Molina, “Exchange Leavitt path algebras and stable rank,” J. Algebr., vol. 305, no. 2, pp. 912–936, 2006, doi: 10.1016/j.jalgebra.2005.12.009.
A. A. Tuganbaev, “Modules over hereditary rings,” Sb. Math., vol. 189, no. 3–4, pp. 623–638, 1998, doi: 10.1070/sm1998v189n04abeh000316.
M. S. Shrikhande, “On Hereditary and Cohereditary Modules,” Can. J. Math., vol. 25, no. 4, pp. 892–896, 1973, doi: 10.4153/cjm-1973-094-2.
G. Zhao and J. Sun, “Global dimensions of rings with respect to a semidualizing module ∗†,” pp. 1–10.
A. U. Ansari and B. K. Sharma, “Graded S-Noetherian Modules,” Int. Electron. J. Algebr., vol. 33, no. 33, pp. 87–108, 2023, doi: 10.24330/ieja.1229782.
U. N. M, C. Lomp, and H. Fern, “A Note on Prime Modules,” vol. 8, no. 1, pp. 31–42, 2000.
I. E. Wijayanti, “Primeness in Category of Modules and Category of Comodules Over Corings,” J. Indones. Math. Soc., vol. 14, no. 1, pp. 13–24, 2012, doi: 10.22342/jims.14.1.58.13-24.
W. A. Ali and N. S. Al Mothafar, “On Quasi-Small Prime Modules,” J. Phys. Conf. Ser., vol. 1818, no. 1, 2021, doi: 10.1088/1742-6596/1818/1/012204.
Irawati, “The Generalization of HNP Ring and Finitely Generated Module over HNP Ring,” Int. J. Algebr., vol. 5, no. 13, pp. 611–626, 2011.
I. Assem, D. Simson, and a Skowronski, “Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory,” London Math. Soc. Student Texts, vol. 1, 2006, [Online]. Available: http://scholar.google.com/scholar?hl=en&btnG=Search&q=intitle:Elements+of+the+Representation+Theory+of+Associative+Algebras#4%5Cnhttp://books.google.com/books?hl=en&lr=&id=SKC79Pt_c9EC&oi=fnd&pg=PA1&dq=Elements+of+the+Representation+Theory+of+Associative+
D. M. Barquero, C. M. González, and I. R. Campos, “Algebraic characterisations of path algebras,” pp. 1–28, 2023, [Online]. Available: http://arxiv.org/abs/2310.10580
G. Abrams and G. Aranda Pino, “The Leavitt Path Algebras of Generalized Cayley Graphs,” Mediterr. J. Math., vol. 13, no. 1, pp. 1–27, 2016, doi: 10.1007/s00009-014-0464-4.
G. A. Pino, E. Pardo, and M. S. Molina, “Prime spectrum and primitive Leavitt path algebras,” Indiana Univ. Math. J., vol. 58, no. 2, pp. 869–890, 2009, doi: 10.1512/iumj.2009.58.3516.
X. W. Chen, “Irreducible representations of Leavitt path algebras,” Forum Math., vol. 27, no. 1, pp. 549–574, 2015, doi: 10.1515/forum-2012-0020.
K. M. Rangaswamy, “On simple modules over Leavitt path algebras,” J. Algebr., vol. 423, pp. 239–258, 2015, doi: 10.1016/j.jalgebra.2014.10.008.
K. M. Rangaswamy, “A Survey of Some of the Recent Developments in Leavitt Path Algebras,” Indian Stat. Inst. Ser., pp. 3–20, 2020, doi: 10.1007/978-981-15-1611-5_1.
G. Abrams, F. Mantese, and A. Tonolo, “Extensions of simple modules over Leavitt path algebras,” J. Algebr., vol. 431, pp. 78–106, 2015, doi: 10.1016/j.jalgebra.2015.01.034.
Risnawita, Irawati, and I. Muchtadi-Alamsyah, “Primeness of Simple Modules Over Path Algebras and Leavitt Path Algebras,” Khayyam J. Math., vol. 7, no. 2, pp. 219–231, 2021, doi: 10.22034/kjm.2021.203331.1578.
Z. Bı̇lgı̇n, M. L. Reyes, and Ü. Tekı̇r, “On right S-Noetherian rings and S-Noetherian modules,” Commun. Algebr., vol. 46, no. 2, pp. 863–869, 2018, doi: 10.1080/00927872.2017.1332199.
Downloads
Submitted
Accepted
Published
Issue
Section
License
Copyright (c) 2024 Risnawita, Fatemeh Savoji, Dwi Devi Wulandari
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with Lattice Journal : Journal of Mathematics Education and Applied agree to the following terms: Authors retain copyright and grant the Lattice Journal : Journal of Mathematics Education and Applied right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY-SA 4.0) that allows others to share (copy and redistribute the material in any medium or format) and adapt (remix, transform, and build upon the material) the work for any purpose, even commercially with an acknowledgement of the work's authorship and initial publication in Lattice Journal : Journal of Mathematics Education and Applied. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in Lattice Journal : Journal of Mathematics Education and Applied. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
