Research Article

Norm-Compatible Fuzzy Subrings and Quotient Rings

Number: 55 June 30, 2026

Norm-Compatible Fuzzy Subrings and Quotient Rings

Abstract

In the study of fuzzy normed rings, classical frameworks frequently employ idempotent t-norms and t-conorms to govern the underlying algebraic operations. This paper investigates norm-induced fuzzy structures, in which the membership degree is defined by a strictly decreasing and continuous function of the underlying seminorm, and demonstrates an algebraic incompatibility. We prove that the direct application of idempotent operators limits the entire structure to bounded elements within the unit ball. This topological collapse prohibits the multiplicative norm expansion required to model unbounded systems, such as general polynomial or matrix rings. To address this, we introduce a norm-compatibility condition based on strict Archimedean t-norms. Building on this foundation, we construct the fuzzy quotient norm using norm-induced subrings and prove the First Isomorphism Theorem for fuzzy normed rings, thereby establishing that the isometric isomorphism holds without restricting the metric properties of the base ring.

Keywords

Fuzzy normed rings, fuzzy subrings, norm-induced membership, Archimedean t-norms, First Isomorphism Theorem

References

  1. L. A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338–353.
  2. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986) 87–96.
  3. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (2) (1982) 133–139.
  4. N. P. Mukherjee, P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information Sciences 34 (3) (1984) 225–239.
  5. J. N. Mordeson, D. S. Malik, Fuzzy commutative algebra, World Scientific, 1998.
  6. D. A. Molodtsov, Soft set theory–First results, Computers and Mathematics with Applications 37 (4-5) (1999) 19–31.
  7. H. Aktaş, N. Çağman, Soft sets and soft groups, Information Sciences 177 (13) (2007) 2726–2735.
  8. A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (3) (1994) 395–399.
  9. A. Emniyet, M. Şahin, Fuzzy normed rings, Symmetry 10 (10) (2018) 515.
  10. N. A. Alhaleem, A. G. Ahmad, Intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals, Mathematics 8 (9) (2020) 1594.
APA
Emniyet, A. (2026). Norm-Compatible Fuzzy Subrings and Quotient Rings. Journal of New Theory, 55, 58-66. https://doi.org/10.53570/jnt.1919557
AMA
1.Emniyet A. Norm-Compatible Fuzzy Subrings and Quotient Rings. JNT. 2026;(55):58-66. doi:10.53570/jnt.1919557
Chicago
Emniyet, Aykut. 2026. “Norm-Compatible Fuzzy Subrings and Quotient Rings”. Journal of New Theory, nos. 55: 58-66. https://doi.org/10.53570/jnt.1919557.
EndNote
Emniyet A (June 1, 2026) Norm-Compatible Fuzzy Subrings and Quotient Rings. Journal of New Theory 55 58–66.
IEEE
[1]A. Emniyet, “Norm-Compatible Fuzzy Subrings and Quotient Rings”, JNT, no. 55, pp. 58–66, June 2026, doi: 10.53570/jnt.1919557.
ISNAD
Emniyet, Aykut. “Norm-Compatible Fuzzy Subrings and Quotient Rings”. Journal of New Theory. 55 (June 1, 2026): 58-66. https://doi.org/10.53570/jnt.1919557.
JAMA
1.Emniyet A. Norm-Compatible Fuzzy Subrings and Quotient Rings. JNT. 2026;:58–66.
MLA
Emniyet, Aykut. “Norm-Compatible Fuzzy Subrings and Quotient Rings”. Journal of New Theory, no. 55, June 2026, pp. 58-66, doi:10.53570/jnt.1919557.
Vancouver
1.Aykut Emniyet. Norm-Compatible Fuzzy Subrings and Quotient Rings. JNT. 2026 Jun. 1;(55):58-66. doi:10.53570/jnt.1919557