The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings

Abstract

This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings ⟨ R ∪ I ⟩ are given when R is C , R , Q , Z or Z n . Secondly, the neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ is introduced and all idempotents in neutrosophic quadruple rings ⟨ C ∪ T ∪ I ∪ F ⟩ , ⟨ R ∪ T ∪ I ∪ F ⟩ , ⟨ Q ∪ T ∪ I ∪ F ⟩ , ⟨ Z ∪ T ∪ I ∪ F ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ are also given. Furthermore, the algorithms for solving the idempotents in ⟨ Z n ∪ I ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ for each nonnegative integer n are provided. Lastly, as a general result, if all idempotents in any ring R are known, then the structure of idempotents in neutrosophic ring ⟨ R ∪ I ⟩ and neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ can be determined.