1 Affiliation : Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria
Email : email@example.com
2 Affiliation : Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria
Email : firstname.lastname@example.org
3 Affiliation : Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
Email : email@example.com
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:
Neutrosophy , reﬁned neutrosophic set , reﬁned neutrosophic group , reﬁned neutrosophic ring
 Agboola,A.A.A.; Akinola,A.D ; Oyebola, O.Y. ” Neutrosophic Rings I”, Int. J. of Math. Comb., vol 4, pp.1-14, 2011.
 Agboola, A.A.A.; Akwu A.O. ; Oyebo,Y.T. ”Neutrosophic Groups and Neutrosopic Subgroups”, Int. J. of Math. Comb., vol 3, pp. 1-9, 2012 .
 Agboola,A.A.A.; Davvaz, B. ”On Neutrosophic Canonical Hypergroups and Neutrosophic Hyperrings”, Neutrosophic Sets and Systems, vol 2, pp. 34-41, 2014.
 Agboola,A.A.A.; Adeleke, E.O.; Akinleye, S.A. ”Neutrosophic Rings II”, Int. J. of Math. Comb., vol 2, pp. 1-8, 2012.
 Agboola,A.A.A. ”On Reﬁned Neutrosophic Algebraic Structures”, Neutrosophic Sets and Systems, vol 10, pp. 99-101, 2015.
 Agboola,A.A.A,; Davvaz,B.; Smarandache,F. ”Neutrosophic Quadruple Hyperstructures”, Annals of Fuzzy Mathematics and Informatics, vol 14 (1), pp. 29-42, 2017.
 Akinleye,S.A; Adeleke,E.O ; Agboola,A.A.A. ”Introduction to Neutrosophic Nearrings”, Annals of Fuzzy Mathematics and Informatics, vol 12 (3), pp. 397-410, 2016.
 Akinleye, S.A; Smarandache,F.; Agboola,A.A.A. ”On Neutrosophic Quadruple Algebraic Structures”, Neutrosophic Sets and Systems, vol 12, pp. 122-126, 2016.
 Smarandache,F. ”A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neu-trosophic Probability”, (3rd edition), American Research Press, Rehoboth,2003,
 Smarandache,F. ”n-Valued Reﬁned Neutrosophic Logic and Its Applications in Physics”, Progress in Physics, USA, vol 4, pp. 143-146, 2013.
 Smarandache,F. ”(T,I,F)- Neutrosophic Structures”, Neutrosophic Sets and Systems, vol 8, pp. 3-10, 2015.
 Vasantha Kandasamy,W.B; Smarandache,F. ”Neutrosophic Rings” Hexis, Phoenix, Arizona, 2006, http://fs.gallup.unm.edu/NeutrosophicRings.pdf