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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.

The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings