On Imperfect Duplets In Some Refined Neutrosophic Rings

Abstract

This paper solves the imperfect duplets problem in refined neutrosophic rings, where it presents the necessary and sufficient conditions for a pair to be an imperfect duplet in any refined neutrosophic ring. Also, this work introduces a full description of the structure of imperfect duplets in numerical refined neutrosophic rings such as refined neutrosophic ring of integers , refined neutrosophic ring of rationales , and refined neutrosophic ring or real numbers.