The Role of Boolean and Pseudo Commutativity in Near Rings

C. Dhivya; D. Radha

doi:https://doi.org/10.14445/22315373/IJMTT-V68I9P506

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 9 | Year 2022 | Article Id. IJMTT-V68I9P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I9P506

The Role of Boolean and Pseudo Commutativity in Near Rings

C. Dhivya, D. Radha

Received	Revised	Accepted	Published
28 Jul 2022	01 Sep 2022	15 Sep 2022	27 Sep 2022

Abstract

In this paper we have discussed about Boolean near rings and pseudo commutative in near rings. Hansen, D.J. and Jiang Luh[6] have stated that if N is a Boolean near ring, then xy=xyx for each x,y ε N. It has been proved in this paper by using the lemma if N is a Boolean near ring, then ab=0 implies ba = b0 for all a,b ε N. We have also generalized this lemma to m≥1 and n≥1. It is obtained that every Boolean near ring with commutativity is pseudo commutative.

Keywords

Boolean, Commutative, Pseudo Commutative, Weak Commutative, Zero symmetric.

References

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

[2] C. Dhivya and D. Radha, "𝛽6 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.

[6] Gunter Pilz,, “Near Rings”, North Holland, Amsterdam, 1983.

[7] D.J. Hansen and Jiang Luh, “Boolean Near Rings and Weak Commutativity,” J. Austral. Math. Soc. (Series A), vol. 47, pp. 103 –107, 1989.

[8] Neal. H. Mccoy, “Theory of Rings”, Macmillan & Co, 1970.

[9] D. Radha, C. Dhivya, M. Vinutha, K. Muthu Maheswari and S.R. Veronica Valli, “A Study on R-Near Ring”, Journal of Emerging Technologies and Innovative Research (Jetir), vol. 6, no. 12, pp. 434 – 439, 2019.

[10] D. Radha and C. Dhivya, "On S-Near Rings and S'-Near Rings with Right Bipotency", Jetir, vol. 6, no. 2, pp. 952 - 1000, 2019.

[11] D. Radha, M. Vinutha and C. Raja Lakshmi, "A Study on Gs-Near Ring,” Journal of Emerging Technologies and Innovative Research (Jetir), vol. 6, no. 2, pp. 2349 - 5162, 2019.

[12] D. Radha, C. Dhivya and S.R. Veronica Valli, "A Study on Quasi Weak Commutative Gamma Near Rings,” Journal of Emerging

Technologies and Innovative Research, vol. 6, no. 6, pp. 2349 - 5162, 2019.

[13] D. Radha, M. Vinutha and C. Dhivya, "On Prime Semi Near Rings", Journal of Emerging Technologies and Innovative Research (Jetir), vol. 6, no. 6, pp. 2349 - 5162, 2019.

[14] D. Radha and C. Dhivya, "Role of 𝛼1 and 𝛼2 Near Ring in Boolean S-Near Ring", International Multidisciplinary Innovative Research Journal, vol. 4, no. 1, pp. 2456 - 4613, 2019.

[15] D. Radha and C. Dhivya, “On Unit 𝐵2 Near Rings,” Studies in Indian Place Names, vol. 40, no. 70, pp. 3413 – 3418, 2020.

[16] D. Radha and C. Dhivya, "A Study on Cm (2,2) Near Ring,” Science, Technology and Development, vol. 10, no. 03, pp. 626 - 633, 2021.

[17] D. Radha, C. Dhivya and S. Shiny, "Some Results on S1-Near Rings", Science, Technology and Development, vol. 10, no. 04, pp. 191 -195, 2021.

[18] D. Radha and C. Dhivya, "On Some Characterizations of Cm' (2,2) Near Ring”, Science, Technology and Development, vol. 10, no. 11, pp. 104 - 111, 2021.

[19] D. Radha and C. Dhivya, "On Some Characterizations of R-Near Rings", in Proc. Icamipr 2, pp. 235 - 240.

[20] D. Radha and C. Dhivya, "The Algebraic Structures of Cm (2,2) Near Ring", in Proc. Icmmmhpcst, pp. 409 - 412.

[21] G. Sugantha and R. Balakrishnan, "𝛽1 Near-Rings,” International Journal of Algebra, vol. 08, no. 01, pp. 1 - 7, 2014.

[22] G. Sugantha and R. Balakrishnan, "𝛽2 Near-Rings,” Ultra Scientist, vol. 26(1)A, pp. 63 - 68, 2014.

[23] S. Uma, R. Balakrishnan, and T. Tamizh Chelvam, “Pseudo Commutative Near-Rings,” Scientia Magna, vol. 6, no. 2, pp. 75-85,

2010.

[24] Yong Uk Cho, “On Semicentral Idempotents in Near Rings,” Applied Mathematical Sciences, vol. 9, no. 77, pp. 3843 – 3846,

2015.

[25] Yong Uk Cho, “Properties of Semicentral Idempotent Near-Rings,” Far East Journal of Mathematical Sciences, vol. 100, no. 3, pp. 463 – 468, 2016.

Citation :

C. Dhivya, D. Radha, "The Role of Boolean and Pseudo Commutativity in Near Rings," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 9, pp. 34-38, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I9P506