Relative essential ideals in $N$-groups

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Tapatee Sahoo
Bijan Davvaz
Harikrishnan Panackal
Babushri Srinivas Kedukodi
Syam Prasad Kuncham


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.

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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.


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.