[Paper Review] Smarandache Non-Associative Rings

W. B. Vasantha Kandasamy|ArXiv.org|Jun 2, 2003

Mathematics and Applications37 references3 citations

TL;DR

This paper introduces and systematically studies Smarandache non-associative rings—non-associative rings that contain a proper subset forming an associative ring under the same operations. It establishes foundational definitions, provides 150 problems, and presents 44 tables, offering a comprehensive algebraic framework for exploring rings with mixed algebraic structures.

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).

Motivation & Objective

• To formalize the concept of Smarandache non-associative rings as a generalization of non-associative rings with embedded associative substructures.
• To investigate the interplay between weak (non-associative) ring structures and stronger (associative) subring structures within the same set.
• To provide a systematic foundation for studying algebraic systems where a single set supports both non-associative and associative operations.
• To offer a collection of 150 problems and 44 tables to support further research and pedagogical use in non-associative algebra.
• To extend the Smarandache notion of 'weak structure with a proper subset having a stronger structure' to non-associative rings.

Proposed method

• Define a non-associative ring as a set R with two binary operations '+' and '.' such that (R, +) is an abelian group and (R, .) is a groupoid.
• Impose distributivity: (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b for all a, b, c in R.
• Introduce the Smarandache condition: a proper subset P ⊂ R must form an associative ring under the same operations.
• Use the framework of general algebraic structures to explore the hierarchy between non-associative and associative properties.
• Present 44 tables to classify and illustrate various types of Smarandache non-associative rings.
• Include 150 problems to explore structural properties, closure, and subring behavior in these systems.

Experimental results

Research questions

• RQ1 What conditions must a non-associative ring satisfy to contain a proper subset that forms an associative ring?
• RQ2 How do the distributive laws interact with the non-associativity of the multiplicative operation in such rings?
• RQ3 What are the structural characteristics of Smarandache non-associative rings compared to standard non-associative or associative rings?
• RQ4 How can the concept of a Smarandache structure be consistently applied to non-associative rings?
• RQ5 What are the implications of having both non-associative and associative substructures within the same ring?

Key findings

• The paper successfully defines and characterizes Smarandache non-associative rings as non-associative rings containing a proper subset that is an associative ring under the same operations.
• It establishes that distributivity holds in the same way as in standard rings, even when associativity fails.
• The existence of such rings is demonstrated through construction and classification, supported by 44 illustrative tables.
• The paper provides 150 problems that explore closure, subring properties, and structural constraints in these rings.
• It confirms that the Smarandache structure concept is applicable and meaningful in the context of non-associative rings.
• The work is published by the American Research Press and made available via arXiv with a DOI, ensuring academic accessibility.