Advancing Fuzzy Algebra: Polynomial Subrings, Zero Divisors, and Related Properties

Authors

  • Dinesh Kute Symbiosis Institute of Technology (SIT ), Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India; Applied Sciences and Humanities, Pimpri Chinchwad College of Engineering, Pune, Maharashtra, India https://orcid.org/0000-0002-7285-0165
  • Arundhati Warke Department of Mathematics, Symbiosis Institute of Technology (SIT ), Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India https://orcid.org/0000-0002-6902-7434
  • Anil Khairnar Department of Mathematics, Abasaheb Garware College, Pune, Maharashtra, India https://orcid.org/0000-0003-2187-6362

DOI:

https://doi.org/10.26713/cma.v16i1.2991

Keywords:

Fuzzy subring, Fuzzy zero divisor, Fuzzy polynomial subring, Fuzzy reduced subring, Fuzzy Armendariz subring

Abstract

This paper aims to expand classical ring theory into the fuzzy context, thereby strengthening the theoretical framework of fuzzy algebra. It introduces various types of fuzzy subrings, including fuzzy polynomial subrings and fuzzy matrix subrings, and explores their properties using the concept of fuzzy zero divisors. A significant contribution of this work is the development of the McCoy theorem for fuzzy polynomial subrings, leading to the definition of fuzzy McCoy subrings. Further, by building upon the notion of fuzzy zero divisors, the paper defines new classes of fuzzy subrings, such as fuzzy Armendariz subrings and fuzzy reduced subrings, and examines their fundamental properties and interconnections.

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References

D. Alghazzwi, A. Ali, A. Almutlg, E. Abo-Tabl and A. A. Azzam, A novel structure of q-rung orthopair fuzzy sets in ring theory, AIMS Mathematics 8(4) (2023), 8365 – 8385, DOI: 10.3934/math.2023422.

A. Razaq and G.Alhamzi, On Pythagorean fuzzy ideals of a classical ring, AIMS Mathematics 8(2) (2023), 4280 – 4303, DOI: 10.3934/math.2023213.

H. Alolaiyan, M. H. Mateen, D. Pamucar, M. K. Mahmmod and F. Arslan, A certain structure of bipolar fuzzy subrings, Symmetry 13(8) (2021), 1397, DOI: 10.3390/sym13081397.

E. P. Armendariz, A note on extensions of Baer and P. P.-rings, Journal of the Australian Mathematical Society 18(4) (1974), 470 – 473, DOI: 10.1017/S1446788700029190.

G. Çuvalcıoglu and S. Tarsuslu (Yılmaz), Isomorphism theorems on intuitionistic fuzzy abstract algebras, Communications in Mathematics and Applications 12(1) (2021), 109 – 126, DOI: 10.26713/cma.v12i1.1446.

M. Gulzar, D. Alghazzawi, M. Mateen and N. Kausar, A certain class of t-intuitionistic fuzzy subgroups, IEEE Access 8 (2020), 163260 – 163268, DOI: http://doi.org/10.1109/ACCESS.2020.3020366.

M. Gulzar, F. Dilawar, D. Alghazzawi and M. Mateen, A note on complex fuzzy subfield, Indonesian Journal of Electrical Engineering and Computer Science 21(2) (2021), 1048 – 1056, DOI: 10.11591/ijeecs.v21.i2.pp1048-1056.

D. Kute, A. Warke, A. Khairnar and L. Sharma, A review of fuzzy subring algebra: Theories and case studies, Nanotechnology Perceptions 20(6) (2024), 3364 – 3383, URL: https://nano-ntp.com/index.php/nano/article/view/3483.

D. Kute, A. Warke, A. Khairnar, and L. Sharma, Fuzzy baer subrings: A fuzzified extension of Baer rings, Advances in Nonlinear Variational Inequalities 28(2) (2025), 399 – 409, URL: https://internationalpubls.com/index.php/anvi/article/view/2352.

W.-J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8(2) (1982), 133 – 139, DOI: 10.1016/0165-0114(82)90003-3.

D. S. Malik and J. N. Mordeson, Fuzzy subfields, Fuzzy Sets and Systems 37(3) (1990), 383 – 388, DOI: 10.1016/0165-0114(90)90034-4.

N. H. McCoy, Remarks on divisors of zero, The American Mathematical Monthly 49(5) (1942), 286 – 295, DOI: 10.2307/2303094.

S. Melliani, I. Bakhadach and L. S. Chadli, Fuzzy rings and fuzzy polynomial rings, in: Homological and Combinatorial Methods in Algebra (SAA 2016), A. Badawi, M. Vedadi, S. Yassemi and A. Yousefian Darani (editors), Springer Proceedings in Mathematics & Statistics, Vol. 228, Springer, Cham., (2018), DOI: 10.1007/978-3-319-74195-6_8.

S. Muthukumaran and B. Anandh, Bipolar valued multi fuzzy normal subnear-ring of a near-ring, Communications in Mathematics and Applications 14(1) (2023), 397 – 403, DOI: 10.26713/cma.v14i1.2000.

A. K. Ray, A note on fuzzy characteristic and fuzzy divisor of zero of a ring, Novi Sad Journal of Mathematics 34(1) (2004), 39 – 45, URL: https://sites.dmi.uns.ac.rs/nsjom/Papers/34_1/nsjom_34_1_039_045.pdf.

M. B. Rege and S. Chhawchharia, Armendariz rings, Proceedings of the Japan Academy Series A Mathematical Sciences 73(1) (1997), 14 – 17, DOI: 10.3792/pjaa.73.14.

A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications 35(3) (1971), 512 – 517, DOI: 10.1016/0022-247X(71)90199-5.

L. A. Zadeh, Fuzzy sets, Information and Control 8(3) (1965), 338 – 353, DOI: 10.1016/S0019-9958(65)90241-X.

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Published

13-08-2025
CITATION

How to Cite

Kute, D., Warke, A., & Khairnar, A. (2025). Advancing Fuzzy Algebra: Polynomial Subrings, Zero Divisors, and Related Properties. Communications in Mathematics and Applications, 16(1), 77–89. https://doi.org/10.26713/cma.v16i1.2991

Issue

Vol. 16 No. 1 (2025)

Section

Research Article

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