Abstract
In complex rings or complex fields, the notion of imaginary element i with i 2 = − 1 or the complex number i is included, while, in the neutrosophic rings, the indeterminate element I where I 2 = I is included. The neutrosophic ring ⟨ R ∪ I ⟩ is also a ring generated by R and I under the operations of R. In this paper we obtain a characterization theorem for a semi-idempotent to be in ⟨ Z p ∪ I ⟩ , the neutrosophic ring of modulo integers, where p a prime. Here, we discuss only about neutrosophic semi-idempotents in these neutrosophic rings. Several interesting properties about them are also derived and some open problems are suggested.
Highlights
According to Gray [1], an element α 6= 0 of a ring R is called a semi-idempotent if and only if α is not in the proper two-sided ideal of R generated by α2 − α, that is α ∈
Semi-idempotents have been studied for group rings, semigroup rings and near rings [2,3,4,5,6,7,8,9]
As the newly introduced notions of neutrosophic triplet groups [17,18] and neutrosophic triplet rings [19], neutrosophic triplets in neutrosophic rings [20] and their relations to neutrosophic refined sets [21,22] depend on idempotents, the relative study of semi-idempotents will be an innovative research for any researcher interested in these fields
Summary
Mathematics 2019, 7, 507 semi-idempotents in these neutrosophic rings. The non-trivial neutrosophic idempotents are I and 1 + 2I. Is a non-trivial semi-idempotent even though 2 + 2I is a unit of R.
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