aspg-24-2725-1791482647 20261008180407 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Journal of Neutrosophic and Fuzzy Systems JNFS 2771-6430 2771-6449 2024 8 1 Warshine Barry University of Debrecen, Department of Mathematical and Computational Science, Debrecen, Hungary Narek Badjajian University of Debrecen, Department of Mathematical and Computational Science, Debrecen, Hungary 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. 2024 31 38 2725 https://creativecommons.org/licenses/by/4.0/ 10.54216/JNFS.080104 https://www.americaspg.com/journal/24/article/2725 https://www.americaspg.com/storage/81714178398.pdf