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.Comment: 154 page

Kandasamy, W. B. Vasantha

Smarandache, Florentin

English

arXiv

In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for which the roots can be indeterminates. Further, the notion of neutrosophic matrix rings is defined in this book. Already these neutrosophic matrixes have been applied and used in the neutrosophic models like neutrosophic cognitive maps (NCMs), neutrosophic relational maps (NRMs) and so on.    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

Kandasamy, W.B. Vasantha

University of New Mexico

NEUTROSOPHIC RINGS

In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for which the roots can be indeterminates. Further, the notion of neutrosophic matrix rings is defined in this book. Already these neutrosophic matrixes have been applied and used in the neutrosophic models like neutrosophic cognitive maps (NCMs), neutrosophic relational maps (NRMs) and so on

Vasantha, Kandasamy

ZENODO

A characterization of f-ring without non-zero nilpotents,

A generalization of strongly regular rings,

A note on additive subgroups of finite rings,

A note on f-group rings without non-zero nilpotents,

A note on F-rings,

A note on group rings which are F-rings,

A note on pre J-group rings,

A note on semi idempotents in group rings,

A note on semigroup rings which are Boolean rings,

A note on semigroup rings which are pre p-rings,

A note on the mod p-envelope of a cyclic group,

A note on the modular group ring of finite p-group,

A note on the modular group ring of the symmetric group Sn,

A note on the modular semigroup ring of a finite idempotent semigroup,

A note on units and semi idempotents elements in commutative group rings,

A Radical Approach to Algebra,

a S-neutrosophic semigroup and 〈K ∪ I〉 the neutrosophic rational field of characteristic zero. Does the S-neutrosophic semigroup neutrosophic ring [〈K ∪ I〉] [〈S ∪ I〉] have maximal neutrosophic ideals? Find zero divisors, units and idempotents in

A Structure theorem for generalized J rings,

A Unifying Field in Logics: Neutrosophic Logic, Preface by Charles Le,

Albena Tchamova, Bulgarian Academy of Sciences,

and Suenago Takashi, Generalizations of von Neuman regular rings and n-like rings,

and sufficient condition for the Sneutrosophic semigroup neutrosophic ring

Artinian Modules over a matrix ring, Infinite length modules,

Basic Neutrosophic and their Applications to Fuzzy and Neutrosophic models, Hexis,

Can a semigroup neutrosophic ring have a principal neutrosophic ideal? Justify your claim.

Can the group neutrosophic ring 〈Z12 ∪ I〉 [S6] have any nontrivial pseudo neutrosophic subring? Justify your claim.

Can we find left units in the semigroup neutrosophic ring

Characterize those pseudo neutrosophic ring with 1 if they exist.

Chinese Quat.

Coloring of group rings,

Commutative Algebra,

Commutativity of rings inherited by the location of Herstein's condition,

Definitions Derived from Neutrosophics,

Direct sum of strongly regular rings and zero rings,

Does the group neutrosophic ring 〈K ∪ I〉 [Sn] have any pseudo neutrosophic subring?

Does the semigroup neutrosophic ring 〈Z6 ∪ I〉[S(3)] have any pseudo neutrosophic subring?

Does their exist a class of pseudo neutrosophic rings which contains 1?

Does their exists a S-neutrosophic semigroup neutrosophic ring

Does their exists non trivial units and neutrosophic units in the group neutrosophic ring 〈Z2 ∪ I〉

Exchange rings having stable range one,

Extensions of clean rings,

f-semigroup rings,

Filial semigroups and semigroup rings,

Find a necessary and sufficient condition for the group neutrosophic ring [〈K ∪ I〉][G] to have pseudo neutrosophic subrings and pseudo neutrosophic ideals.

Find a necessary and sufficient condition for the semigroup neutrosophic ring

Find a necessary and sufficient condition for the semigroup neutrosophic ring 〈K ∪ I〉 [G] to be a.

Find a necessary and sufficient condition for the Sneutrosophic semigroup neutrosophic ring

Find a necessary and sufficient condition for the Sneutrosophic semigroup ring to be semisimple.

Find a necessary and sufficient condition on

Find condition on 〈K ∪

Find the necessary and sufficient condition for the Sneutrosophic semigroup ring to be a.

Finite rings which has isomorphic quotient rings formed by non-maximal ideals,

Finite zeros and finite zerodivisors,

Group rings which are a direct sum of subrings,

Group rings which have trivial subrings,

Group rings which satisfy super ore condition,

Guassian ideals and the Dedekind- Merlens lemma,

Higher level orders on non-commutative rings,

I〉 [S(n) ] be semisimple?

I〉 [S(p)] be semisimple? a. Minimal neutrosophic ideal? b. Principal neutrosophic ideal? c. Prime neutrosophic ideal?

Ideal theory,

Idempotents in the group ring of a cyclic group,

Infinite Group Rings, Pure and Appl.

Inner associative rings,

Inner Commutative Rings, Sictiuan Daxue Xuebao,

Inner Zero Square ring,

Introduction to Commutative Algebra,

Is the S-neutrosophic semigroup ring

Modules with a group of operators,

Monic and Monic free ideals in a polynomial semiring,

Multi structures and Multi spaces,

n-ideal rings,

Neutrosophic Logic— Generalization of the Intuitionistic Fuzzy Logic, Special Session on Intuitionistic Fuzzy Sets and Related Concepts,

Nil generalized Hamiltonian rings, Heilongiang Daxue Ziran Kexue Xuebao,

Obedient ideals in a finite ring,

Obtain some interesting properties about S-neutrosophic semigroup rings

Obtain some interesting properties about the S-neutrosophic semigroup neutrosophic rings

Neutrosophic Rings

Abstract

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University of New Mexico

ZENODO