Relative essential ideals in $N$-groups

Article Sidebar

PDF
Published: Feb 1, 2023
DOI: https://doi.org/10.5556/j.tkjm.54.2023.4136
Keywords:
Nearring N-groups essential ideal direct summand complement

Main Article Content

Tapatee Sahoo
Department of Mathematics, Manipal institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India
Bijan Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Harikrishnan Panackal
Department of Mathematics, Manipal institute of Technology, Mani- pal Academy of Higher Education, Manipal, Karnataka, India
Babushri Srinivas Kedukodi
Department of Mathematics, Manipal institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India
Syam Prasad Kuncham
Department of Mathematics, Manipal institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India

Abstract

Let $G$ be an $N$-group where $N$ is a (right) nearring. We introduce the concept of relative essential ideal (or $N$-subgroup) as a generalization of the concept of essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or $N$-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of $N$-groups and obtain their properties under homomorphism.

Article Details

How to Cite
Sahoo, T., Davvaz, B., Panackal, H., Kedukodi, B. S., & Kuncham, S. P. (2023). Relative essential ideals in $N$-groups. Tamkang Journal of Mathematics, 54(1), 69–82. https://doi.org/10.5556/j.tkjm.54.2023.4136
Issue
Vol. 54 No. 1 (2023)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

E. Aichinger, F. Binder, F. Ecker, P. Mayr and C. Nöbauer, SONATA-system of near-rings and their applications, GAP package, Version 2.8; 2015.

F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Springer-Verlag New York, 13, 1992.

H. Nayak, S. P. Kuncham, B. S. Kedukodi, ΘΓ N-group, Matematicki Vesnik,70(1)(2018), 64-78.

J. D. P. Meldrum, Near-rings and their links with groups, Pitman Advanced Pub. Program, Vol., 134 1985.

A. Oswald, Near-rings in which every N-subgroup is principal, Proceedings of the London Mathematical Society, 3(1) (1974), 67-88.

A. Oswald, Completely reducible near-rings, Proceedings of the Edinburgh Mathematical Society, 20(3) (1977), 187-197.

G. Pilz, Near-Rings:the theory and its applications, 23, NorthHolland, 1983.

Y. V. Reddy, S. Bhavanari, A Note on N-Groups, I Indian J. Pure Appl. Math., 19(9)(1988), 842-845.

S. Bhavanari, On modules with finite Goldie Dimension, J. Ramanujan Math. Soc., 5(1) (1990) 61-75.

S.Bhavanari and S. P. Kuncham, On direct and inverse systems in N-groups, Indian J. Math., 42(2) (2000) 183-192.

S. Bhavanari and S.P. Kuncham, Linearly Independent Elements in N-groups with Finite Goldie Dimension, Bull. Korean Math. Soc., 42(3) (2005), 433-441.

S. Bhavanari, Goldie dimension and spanning dimension in modules and N-groups, Near- rings, Nearfields and Related topics (Review Volume), World Scientific Publ. Co., (2017) 26-41.

S. Bhavanari, Contributions to near-ring theory, Doctrol Thesis, Nagarjuna University, India, 1984.

S. Bhavanari and S.P. Kuncham, On Finite Goldie Dimension of Mn(N)-group Nn, Proc. of the Nearrings and Nearfields (Springer), (2005), 301-310.

S. Bhavanari and S.P. Kuncham, A Result on E-direct Systems in N-groups, Indian J. pure appl. Math., 29(3) (1998) 285-287.

S. Bhavanari and S. P. Kuncham, Near rings, fuzzy ideals, and graph theory. CRC press, 2013.

S. P. Kuncham, Contributions to near-ring theory II. Diss. Doctoral Thesis, Nagarjuna University, 2000.

S. Bhavanari, S. P. Kuncham, V. R. Paruchuri and B. Mallikarjuna, A Note on dimensions in N-groups, Italian Journal of Pure and Applied Mathematics, 44 (2020) 649-657.