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Neutrosophic Rings

By Smarandache, Florentin

Book Id:WPLBN0002828478 Format Type:PDF (eBook) File Size:593.86 kb Reproduction Date:8/2/2013

Smarandache, F., & Vasantha Kandasamy, W. B. (2013). Neutrosophic Rings. Retrieved from http://gutenberg.us/

Description
This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings, neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem.

Summary
In this book we define the new notion of neutrosophic rings.

Excerpt
Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group.
DEFINITION 1.1.2: Let N(G) = 〈G ∪ I〉 be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a neutrosophic subgroup if P(G) is a neutrosophic group i.e. P(G) must contain a (sub) group.
Example 1.1.3: Let N(Z2) = 〈Z2 ∪ I〉 be a neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a neutrosophic group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo neutrosophic groups (subgroups).
We can thus define a pseudo neutrosophic group as a neutrosophic group, which does not contain a proper subset which is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different.
Now we see a neutrosophic group can have substructures which are pseudo neutrosophic groups which is evident from the following example.

Table of Contents
Preface 5
Chapter One
INTRODUCTION
1.1 Neutrosophic Groups and their Properties 7
1.2 Neutrosophic Semigroups 20
1.3 Neutrosophic Fields 27
Chapter Two
NEUTROSOPHIC RINGS AND THEIR PROPERTIES
2.1 Neutrosophic Rings and their Substructures 29
2.2 Special Type of Neutrosophic Rings 41
Chapter Three
NEUTROSOPHIC GROUP RINGS AND THEIR GENERALIZATIONS
3.1 Neutrosophic Group Rings 59
3.2 Some special properties of Neutrosophic Group Rings 73
3.3 Neutrosophic Semigroup Rings and their Generalizations 85
Chapter Four
SUGGESTED PROBLEMS 107
REFERENCES 135
INDEX 149
ABOUT THE AUTHORS 154