Many new results were obtained in this paper about P.P. ring. Semi-primitive ring R with (dcc) on principal-ideal P is always P.P.ring. Also, St-G-P.P. ring R is given to answer new question; is B-rings are P.P. ring. Also regular and Von Neumann regular rings are introduced to find a relationship between P.P. ring and these kind of the rings.
Interval complex neutrosophic soft sets (I-CNSSs) are interval neutrosophic soft sets (I-NSSs) described by three two-dimensional independent membership functions which are uncertainty interval, indeterminacy interval, and falsity interval respectively. Relation is a tool that helps in describing consistency and agreement between objects. Throughout this paper, we insert and discuss the interval complex neutrosophic soft relation (simply denoted by I-CNSR) that is a novel soft computing technique used to examine the degree of interaction between two corresponding models called I-CNSSs. We present the definition of the Cartesian product of I-CNSSs followed by the definition of I-CNSR. Further, the definitions and some theorems and properties related to the composition, inverse, and complement of I-CNSR are provided. The notions of symmetric, reflexive, transitive, and equivalent of I-CNSRs are proposed and the algebraic properties of these concepts are verified. Additionally, we point the contribution of our concept to real life problems by presenting a proposed algorithm to solve a real-life decision-making problem. Finally, a comparison between the proposed model and the existing relations is conducted to clarify the importance of this model.
Interval complex neutrosophic soft set (ICNSS) is the generalization of complex neutrosophic soft set (CNSS) as it provides an interval-based membership structure to handle the complex neutrosophic soft data. However, in the definition of the ICNSS, parameters set is a classical set, and the parameters have the same degree of importance which is considered as 1. This poses a limitation in modeling of some problems. Therefore, we introduce the concept of fuzzy parameterized interval complex neutrosophic soft set (FP-ICNSS) based on idea that each of elements of parameters set has got an importance degree. The basic theoretical operations and properties are defined and verified on FP-ICNSS. For FP-ICNSS, we conceptualize the relevant mapping and study the properties of the FP-ICNSS images and inverse images. Then, we propose a new algorithm that is applicable in the field of medical diagnosis and decision-making problems for selection right product. Moreover, an illustrative example is presented which depicts its validity for successful application to the problems involving vagueness and uncertainties. Eventually, a comparison between the proposed model and the existing methods is conducted to clarify the importance of this model.
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