International Journal of Neutrosophic Science

The objective of this paper is to present the concept of a refined neutrosophic vector space. Weak(strong) refined neutrosophic vector spaces and subspaces, and, strong refined neutrosophic quotient vector spaces are studied. Several interesting results and examples are presented. It is shown that every weak (strong) refined neutrosophic vector space is a vector space and it is equally shown that every strong refined neutrosophic vector space is a weak refined neutrosophic vector space.

This article introduces the concept of AH-submodule, AHS-submodule of a neutrosophic module, and AHS-homomorphism. This work presents some basic notions and properties of these concepts such as AH-Kernel, AH-Quotient, and dimension, and determines the algebraic structure of weak neutrosophic module over a commutative ring R. 

This paper studies classical homomorphisms between n-refined neutrosophic ring and m-refined neutrosophic ring. It proves that every m-refined neutrosophic ring  is a homomorphic image of n-refined neutrosophic ring , where . Also, it presents a discussion of kernels and some corresponding isomorphisms between those rings.

The objective of this paper is to introduce the concept of NeutroRings by considering three NeutroAxioms (NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication over addition)). Several interesting results and examples on NeutroRings, NeutroSubgrings, NeutroIdeals, NeutroQuotientRings and NeutroRingHomomorphisms are presented. It is shown that the 1st isomorphism theorem of the classical rings holds in the class of NeutroRings.

Modules are one of the fundamental and rich algebraic structure concerning some binary operations in the study of algebra. In this paper, some basic structures of refined neutrosophic R-modules and refined neutrosophic submodules in algebra are generalized. Some properties of refined neutrosophic R-modules and refined neutrosophic submodules are presented. More precisely, classical modules and refined neutrosophic rings are utilized. Consequently, refinedneutrosophic R- modules that are completely different from the classical modular in the structural properties are introduced. Also, neutrosophic R-module homomorphism is explained and some definitions and theorems are presented.