## Smarandache Non-Associative (SNA-) rings

**Authors:** W. B. Vasantha Kandasamy

In this paper we introduce the concept of Smarandache non-associative rings,
which we shortly denote as SNA-rings as derived from the general definition of a
Smarandache Structure (i.e., a set A embedded with a week structure W such that a
proper subset B in A is embedded with a stronger structure S). Till date the concept of
SNA-rings are not studied or introduced in the Smarandache algebraic literature. The
only non-associative structures found in Smarandache algebraic notions so far are
Smarandache groupoids and Smarandache loops introduced in 2001 and 2002. But they
are algebraic structures with only a single binary operation defined on them that is nonassociative.
But SNA-rings are non-associative structures on which are defined two
binary operations one associative and other being non-associative and addition distributes
over multiplication both from the right and left. Further to understand the concept of
SNA-rings one should be well versed with the concept of group rings, semigroup rings,
loop rings and groupoid rings. The notion of groupoid rings is new and has been
introduced in this paper. This concept of groupoid rings can alone provide examples of
SNA-rings without unit since all other rings happens to be either associative or nonassociative
rings with unit. We define SNA subrings, SNA ideals, SNA Moufang rings,
SNA Bol rings, SNA commutative rings, SNA non-commutative rings and SNA
alternative rings. Examples are given of each of these structures and some open problems
are suggested at the end.

**Comments:** 13 pages

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### Submission history

[v1] 10 Mar 2010

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