Smarandache Near-rings

TL;DR

This paper introduces the concept of Smarandache Near-rings, a structure where a near-ring contains a proper subset that forms a near-field, expanding the understanding of algebraic structures with embedded stronger substructures.

Contribution

It defines Smarandache Near-rings as near-rings containing a proper near-field subset, providing a new perspective on algebraic structures with embedded stronger properties.

Findings

Smarandache Near-rings generalize existing near-ring concepts.

Existence of proper near-field subsets within near-rings.

Potential applications in algebraic structure classification.

Abstract

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B contained in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Near-ring is a non-empty set N together with two binary operations '+' and '.' such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c belonging to N we have (a + b) . c = a . c + b . c A Near-field is a non-empty set P together with two binary operations '+' and '.' such that (P, +) is a group (not-necessarily abelian), {P\{0}, .) is a group. For all a, b, c belonging to P we have (a + b) . c = a . c + b . c A Smarandache Near-ring is a near-ring N which has a proper subset P contained in N, where P is a near-field (with respect to the same binary operations on N).