The neutrosophic triplets in neutrosophic rings Q ∪ I and R ∪ I are investigated in this paper. However, non-trivial neutrosophic triplets are not found in Z ∪ I . In the neutrosophic ring of integers Z \ {0, 1}, no element has inverse in Z. It is proved that these rings can contain only three types of neutrosophic triplets, these collections are distinct, and these collections form a torsion free abelian group as triplets under component wise product. However, these collections are not even closed under component wise addition.

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.

In this paper we study the neutrosophic triplet groups for a ∈ Z 2p and prove this collection of triplets (a, neut(a), anti(a)) if trivial forms a semigroup under product, and semi-neutrosophic triplets are included in that collection. Otherwise, they form a group under product, and it is of order (p − 1), with (p + 1, p + 1, p + 1) as the multiplicative identity. The new notion of pseudo primitive element is introduced in Z 2p analogous to primitive elements in Z p , where p is a prime. Open problems based on the pseudo primitive elements are proposed. Here, we restrict our study to Z 2p and take only the usual product modulo 2p.

In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, Z p ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section.