Smarandache Non-Associative Rings

TL;DR

This paper explores Smarandache non-associative rings, which are non-associative rings containing a proper subset that forms an associative ring, highlighting their structural properties and potential applications.

Contribution

It introduces the concept of Smarandache non-associative rings and characterizes their structure, expanding the understanding of algebraic systems with embedded stronger structures.

Findings

Identification of conditions for subsets to form associative rings within non-associative rings

Examples illustrating Smarandache non-associative rings

Theoretical properties of these rings and their subsets

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 non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c belonging to R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b. A Smarandache non-associative ring is a non-associative ring (R, +, .) which has a proper subset P contained in R, that is an associative ring (with respect to the same binary operations on R).