Computing Idempotent Elements In 3-Cyclic and 4-Cyclic Refined Neutrosophic Rings of Integers

Abstract

An element X in a ring R is called idempotent if it equals its square. In this paper, we study the idempotent elements in the 3-cyclic refined neutrosophic ring of integers and 4-cyclic refined neutrosophic ring of integers, where we compute all idempotents in those two rings by solving many different linear Diophantine systems which are generated directly from their the algebraic structure. On the other hand, we use the same Diophantine systems to compute all 2-potent 3-cyclic, and 4-cyclic refined neutrosophic integer elements.