On uniform ideals and finite Goldie dimension in $R$-groups


Abstract views: 10 / PDF downloads: 11

Authors

  • Tapatee Sahoo Manipal Institute of Technology Bengaluru, Manipal Academy of Higher Education
  • Harikrishnan Panackal Manipal Institute of Technology, Manipal Academy of Higher Education
  • Babushri Srinivas Kedukodi Manipal Institute of Technology, Manipal Academy of Higher Education
  • S P Kuncham Manipal Institute of Technology, Manipal Academy of Higher Education

Keywords:

Nearring, essential ideal, uniform ideal, finite dimension

Abstract

We consider an $R$-group $G$, where $R$ is a (right) nearring.  We introduce the notions relative uniform and strictly relative uniform ideals (or $R$-subgroup) which are not uniform, in general.  We prove important properties and obtain a characterization for an $R$-subgroup to have finite Goldie dimension, in terms of strictly relative uniform $R$-subgroups. We provide the necessary  examples.

Author Biographies

Tapatee Sahoo, Manipal Institute of Technology Bengaluru, Manipal Academy of Higher Education

Department of Mathematics, Manipal Institute of Technology Bengaluru,
Manipal Academy of Higher Education, Institute of Eminence Deemed to be University           Bengaluru, Karnataka, India.

Harikrishnan Panackal, Manipal Institute of Technology, Manipal Academy of Higher Education

Department of Mathematics, Manipal Institute of Technology,
Manipal Academy of Higher Education, Institute of Eminence Deemed to be University           Manipal - 576104, Karnataka, India.

Babushri Srinivas Kedukodi, Manipal Institute of Technology, Manipal Academy of Higher Education

Department of Mathematics, Manipal Institute of Technology,
Manipal Academy of Higher Education, Institute of Eminence Deemed to be University           Manipal - 576104,  Karnataka, India.

References

Anderson F. W, Fuller, K. R, Rings and categories of modules, Graduate Texts in Mathematics, Springer-Verlag

New York, 13, 1992.

Badawi A, On weakly semiprime ideals of commutative rings. Beitr Algebra Geom 57, 589–597, 2016.

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

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

Bhavanari S. and Guntupalli K.R., On a class of modules and N-groups. J. of Indian Math. Soc, 59, 39-44,

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

Bhavanari S. and Kuncham S.P., On Finite Goldie Dimension of Mn(N)-Group Nn, In: Proceedings of the

Conference on Nearrings and Nearfields, July 27–August 3, 2003, Hamburg, Germany.

Bhavanari S., and Kuncham S.P., Linearly independent elements in N-groups with Finite Goldie Dimension,

Bull. Korean Math. Soc., 42(3), 433-441, 2005.

Bhavanari S., Kuncham S.P., On finite Goldie dimension of Mn(N)-group Nn, Nearrings and

Nearfields,(Springer) 301-310, 2005.

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

Bhavanari S., Kuncham S.P., Paruchuri V.R., and Mallikarjuna B., A note on dimensions in R-groups, Italian

Journal of Pure and Applied Mathematics, 44, 649-657, 2020.

Booth, G.L. and Groenewald, N.J., On primeness in matrix near-rings. Arch. Math., 56(6), 539-546, 1991.

Davoudian, M., 2019. On Pseudo-uniform modules, Palest. J. Math., 8(2), 15-21, 2019.

Goldie A.W., The structure of noetherian rings, Lectures on Rings and Modules, vol 246. Springer, Berlin,

Heidelberg, 1972.

Juglal, S., Groenewald, N. J., and Lee, K. S. E., Different prime R-ideals, Algebra Colloq., 17(1), 887-904, 2010.

Meldrum, J.D.P., Van der Walt, A.P.J.,Matrix near-rings, Arch. Math. 47(4), 312–319 (1986)

Meldrum, J.D.P., Near-rings and their links with groups, Bull. Amer. Math. Soc., 17, 156-160, 1985.

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

Nayak Hamsa, Kedukodi B.S, and Kuncham S.P., On central Boolean rings and Boolean type fuzzy ideals,

Kuwait J. Sci., 46(4), 23-32, 2019.

Oswald A., Near-rings in which every N-subgroup is principal, Proc. Lond. Math. Soc., 3(1), 67-88, 1974.

Pilz Gunter, Near-Rings: the theory and its applications, 23, North Holland, 1983.

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

Tapatee S, Kedukodi BS, Shum KP, Harikrishnan PK, Kuncham SP (2021) On essential elements in a lattice

and Goldie analogue theorem. Asian-Eur. J. Math. 2250091. https://doi.org/10.1142/S1793557122500917

Tapatee S., Kedukodi B.S., Shum K.P., Harikrishnan P.K., Kuncham S.P., On the finite Goldie dimension of

sum of two ideals of an R-group, Discussiones Mathematicae: General Algebra & Applications, 43(2), 177–187,

Tapatee S., Davvaz B., Harikrishnan P.K., Kedukodi B.S., Kuncham S.P., Relative Essential Ideals in R-groups,

Tamkang J. Math., 54, 69–82, 2023. https://doi.org/10.5556/j.tkjm.54.2023.4136

Tapatee S., Kedukodi B.S., Juglal S., Harikrishnan P.K., Kuncham S.P.: Generalization of prime ideals

in Mn(N)-group Nn, Rendiconti del Circolo Matematico di Palermo Series 2, 72(1), 449–465, 2023.

https://doi.org/10.1007/s12215-021-00682-y

Tapatee S., Meyer J.H., Harikrishnan P.K., Kedukodi B.S., Kuncham S.P., Partial Order in Matrix Nearrings.

Bull. Iranian Math. Soc., 48(6), 3195–3209, 2022. https://doi.org/10.1007/s41980-022-00689-w

Rajani, S., Tapatee, S., Harikrishnan, P.K., Kedukodi, B.S., Kuncham, S.P., Superfluous ideals

of N-groups, Rendiconti del Circolo Matematico di Palermo Series 2, 72(8), 4149–4167, 2023.

https://doi.org/10.1007/s12215-023-00888-2

Tapatee S., Harikrishnan P.K., Kedukodi B.S., Kuncham S.P., Graph with respect to superfluous elements in

a lattice, Miskolc Mathematical Notes, 23(2), 929-945, 2022. http://dx.doi.org/10.18514/MMN.2022.3620

Downloads

Published

2024-11-18

How to Cite

Sahoo, T., Panackal, H., Kedukodi, B. S., & Kuncham, S. P. (2024). On uniform ideals and finite Goldie dimension in $R$-groups. Results in Nonlinear Analysis, 7(4), 52–63. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/416