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Some characterizations of Smarandache special definite groups

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Abstract

In this paper, we study Smarandache special definite groups. We give necessary and sufficient conditions for a group to be Smarandache special definite group(S-special definite group). Moreover we study Smarandache special definite subgroups, Smarandache special definite maximal ideals, Smarandache special definite minmal ideals.

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New notions are introduced in algebra in order to better study the congruences in number theory.
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In this book, we introduce the notion of Smarandache special definite algebraic structures. We can also call them equivalently as Smarandache definite special algebraic structures. These new structures are defined as those strong algebraic structures which have in them a proper subset which is a weak algebraic structure. For instance, the existence of a semigroup in a group or a semifield in a field or a semiring in a ring. It is interesting to note that these concepts cannot be defined when the algebraic structure has finite cardinality i.e., when the algebraic structure has finite number of elements in it. This book has four chapters. Chapter one is introductory in nature. In chapter two, the notion of Smarandache special definite groups and Smarandache special definite fields are introduced and several interesting properties are derived. The notion of Smarandache definite special rings, vector spaces and linear algebras are introduced and analyzed in chapter three. The final chapter suggests over 200 problems.
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