Modul Herediter Noether dan Prima (HNP) Dari Modul Sederhana Chen A_`

Authors

  • Risnawita Universitas Islam Negeri Sjech M Djamil Djambek Bukittinggi, Indonesia
  • Fatemeh Savoji Urmia University, Tehran, Iran, Islamic Republic of
  • Dwi Devi Wulandari Universitas Islam Negeri Sjech M Djamil Djambek Bukittinggi, Indonesia

DOI:

https://doi.org/10.30983/lattice.v4i1.8679

Keywords:

Herediter, Noether, Prima

Abstract

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.

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Published

2024-06-30

How to Cite

Risnawita, Savoji, F., & Dwi Devi Wulandari. (2024). Modul Herediter Noether dan Prima (HNP) Dari Modul Sederhana Chen A_`. Lattice Journal : Journal of Mathematics Education and Applied, 4(1), 77–89. https://doi.org/10.30983/lattice.v4i1.8679

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Vol. 4 No. 1 (2024): Juni 2024

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