A Systematic Inquiry of Algebraic Implications of Boolean–Like Near Rings Under Pseudo Commutativity

Authors

  • S. R. Shimony Rathna Kumari Research Scholar, PG and Research Department of Mathematics, A. P. C. Mahalaxmi College for Women, Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu, India. Author
  • R. Rajeswari Assistant Professor, PG and Research Department of Mathematics, A. P. C. Mahalaxmi College for Women, Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu, India. Author

DOI:

https://doi.org/10.66566/ijmir/2026.v6n3.07

Keywords:

Near-ring, Boolean-like Near Ring, Pseudo Commutativity, Distributivity, Reduced, Nilpotent, Idempotent.

Abstract

A. L.Foster and Alfred L.Foster introduced “Boolean-like rings” as generalizations of Boolean rings, and this concept was extended to “Boolean-like near-rings” by researchers like Clay, James.R and Lawver, Donald. In this paper, we have established conditions under which pseudo-commutative Boolean-like near-rings exhibit properties such as idempotency, distributive behaviour, and reduced ideal structures. Pseudo-commutativity is shown to impose significant constraints on the multiplication structure, leading to results analogous to those obtained under full commutativity. We defined the equivalence relation N on R×S by (????1, ????1) ~ (????2, ????1) if there exists an element ???? ∈ ???? such that ????(????1????2 − ????2????1) = 0. The study demonstrates that pseudo-commutativity serves as an effective substitute for commutativity in deriving key structural results, thereby broadening the scope of Boolean-like near-ring theory and offering new directions for further research.

References

[1] R. Balakrishnan and S. Suryanarayanan, “On ???????? and ???????? ′ Near – Rings”, Bull. Malaysian Math., Sc. Soc. (Second Series), vol. 23, pp. 9 – 24, 2000.

[2] C. Dhivya and D. Radha, "????6 Near Rings", Science, Technology and Development, vol. 11, no. 05, pp. 438 - 443, 2022.

[3] S. Geetha and G. Gopalakrishnamoorthy, “On Quasi – Weak Commutative Near – Rings Iii”, Advances in Mathematics: Scientific Journal, vol. 8, no. 3, pp. 423 – 429, 2019.

[4] G. Gopalakrishnamoorthy, M. Kamaraj Ands. Geetha, “On Quasi Weak Commutative Near-Rings”, International Journal of Mathematics Research, vol. 5, no. 5, pp. 431-440, 2013.

[5] Gratzer and George, “Universal Algebra”, Van Nostrand, 1968.

Cover Page

Published

01-07-2026

Issue

Vol. 6 No. 3 (2026)

Section

Articles

License

Copyright (c) 2026 Council of Industrial Innovation and Research (CIIR), Noida, India.

Creative Commons License

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

How to Cite

[1]
S. R. Shimony Rathna Kumari and R. Rajeswari, “A Systematic Inquiry of Algebraic Implications of Boolean–Like Near Rings Under Pseudo Commutativity”, Int. J. Multidiscip. Innovat. Res., vol. 6, no. 3, pp. 63–69, Jul. 2026, doi: 10.66566/ijmir/2026.v6n3.07.