Algebraic Structure of Neutrosophic Duplets in Neutrosophic Rings , Q∪I, and

Abstract

The concept of neutrosophy and indeterminacyIwas introduced by Smarandache, to deal with neutralies.Since then the notions of neutrosophic rings, neutrosophic semigroups and other algebraic structures have been de-veloped. Neutrosophic duplets and their properties were introduced by Florentin and other researchers have pursuedthis study.In this paper authors determine the neutrosophic duplets in neutrosophic rings of characteristic zero. Theneutrosophic duplets of〈Z∪I〉,〈Q∪I〉and〈R∪I〉; the neutrosophic ring of integers, neutrosophic ring of rationalsand neutrosophic ring of reals respectively have been analysed. It is proved the collection of neutrosophic dupletshappens to be infinite in number in these neutrosophic rings. Further the collection enjoys a nice algebraic structurelike a neutrosophic subring, in case of the duplets collection{a−aI|a∈Z}for which1−Iacts as the neutral. For theother type of neutrosophic duplet pairs{a−aI,1−dI}wherea∈R+andd∈R, this collection under componentwise multiplication forms a neutrosophic semigroup. Several other interesting algebraic properties enjoyed by themare obtained in this paper.