Study of Fuzzy σ-Ring  and Some Related Concepts

Main Article Content

Ibrahim S Ahmed
hassan H. Ebrahim
Ali Al-Fayadh


This paper introduces the concept of fuzzy σ-ring as a generalization of fuzzy σ-algebra and basic properties; examples of this concept have been given.  As the first result, it has been proved that every  σ-algebra over a fuzzy set x*  is a  fuzzy σ-ring-over a fuzzy set x*  and construct their converse by example. Furthermore, the fuzzy ring  concept has been studied to generalize fuzzy algebra and its relation. Investigating  that the concept of fuzzy  σ-Ring is a stronger form of a fuzzy ring  that is every fuzzy σ-Ring over a fuzzy set x* is a fuzzy ring over a fuzzy set x* and construct their converse by example. In addition, the idea of the smallest, as an important property in the study of real analysis, is studied as well. Finally, the main goal of this paper is to study these concepts and give basic properties, examples, characterizations and relationships between them.

Article Details

How to Cite
Ahmed, I. S., Ebrahim, hassan H., & Al-Fayadh, A. (2022). Study of Fuzzy σ-Ring  and Some Related Concepts. Ibn AL- Haitham Journal For Pure and Applied Sciences, 35(2), 37–46.


Ahmed, I.S. ; Ebrahim, H.H. Generalizations of σ-field and new collections of sets noted by δ-field, AIP Conf Proc. 2019, 2096, (020019-1 )-(020019-6).

Ahmed, I.S. ; Asaad, S.H. ; Ebrahim, H.H. Some new properties of an outer measure on a σ–field, Journal of Interdisciplinary Mathematics. 2021, 24 (4), 947–952.

Endou, N. ; Nakasho, K. ; Shidama, Y. σ-ring and σ-algebra of Sets, Formaliz. Math. 2015, 23 (1), 51–57.

Ahmed, I.S. ; Ebrahim, H.H. On α-field and β-field, J. Phys.: Conf. Ser. 2019, 1294, 1-8.

Ebrahim, H.H.; Ahmed, I.S. On a New Kind of Collection of Subsets Noted by δ–field and Some Concepts Defined on δ–field, Ibn Al Haitham Journal for Pure and Applied Science. 2019, 32 (2) , 62-70.

Ebrahim, H.H.; Rusul, A.A. λ–Algebra With Some Of Their Properties, Ibn Al Haitham Journal for Pure and Applied Science. 2020, 33 (2) ,72-80.

Zadeh, L. Fuzzy Sets, Information and Control. 1965, 8, 338–353.

Brown, J.G. A note on fuzzy sets, Inf. Control. 1971, 18 (1), 32–39.

Wang, Z. ; Klir, G.J. Fuzzy Measure Theory; 1st ed.; Springer Science and Business Media, LLC, New York, 1995; ISBN 978-1-4419-3225-10.Ahmed, I.S. ; Ebrahim, H.H. ; Al-Fayadh, A. Fuzzy σ–algebra and some related concepts, Journal of Physics: Conference Series, that will come out in January 2022.