Volume 2 • Issue 2 • PP: 77-81 • 2020
Refined Neutrosophic Rings I
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© 2020 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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The notion of neutrosophic ring R(I) generated by the ring R and the indeterminacy component I was introduced for the first time in the literature by Vasantha Kandasamy and Smarandache in.12 Since then, fur-ther studies have been carried out on neutrosophic ring, neutrosophic nearring and neutrosophic hyperring see.1, 3, 4, 6–8 Recently, Smarandache10 introduced the notion of refined neutrosophic logic and neutrosophic set with the splitting of the neutrosophic components < T, I, F > into the form
< T1, T2, . . . , Tp; I1, I2, . . . , Ir; F1, F2, . . . , Fs > where Ti, Ii, Fi can be made to represent different logical notions and concepts. In,11 Smarandache introduced refined neutrosophic numbers in the form (a, b1I1, b2I2, . . . , bnIn) where a, b1, b2, . . . , bn ∈ R or C. The concept of refined neutrosophic algebraic structures was introduced by Agboola in5 and in particular, refined neutrosophic groups and their substructures were studied. The present paper is devoted to the study of refined neutrosophic rings and their substructures. It is shown that every refined neutrosophic ring is a ring.
For the purposes of this paper, it will be assumed that I splits into two indeterminacies I1 [contradiction (true (T) and false (F))] and I2 [ignorance (true (T) or false (F))]. It then follows logically that:
Keywords
References
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[9] Smarandache,F. ”A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neu-trosophic Probability”, (3rd edition), American Research Press, Rehoboth,2003,
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[10] Smarandache,F. ”n-Valued Refined Neutrosophic Logic and Its Applications in Physics”, Progress in Physics, USA, vol 4, pp. 143-146, 2013.
[11] Smarandache,F. ”(T,I,F)- Neutrosophic Structures”, Neutrosophic Sets and Systems, vol 8, pp. 3-10, 2015.
[12] Vasantha Kandasamy,W.B; Smarandache,F. ”Neutrosophic Rings” Hexis, Phoenix, Arizona, 2006, http://fs.gallup.unm.edu/NeutrosophicRings.pdf
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