## Smarandache Idempotents in Loop Rings Z_{t}L_{n}(m) of the Loops L_{n}(m):

**Authors:** W.B.Vasantha, Moon K. Chetry

In this paper we establish the existance of S-idempotents in case of loop rings
Z_{t}L_{n}(m) for a special class of loops L_{n}(m); over the ring of modulo integers
Z_{t} for a specific value of t. These loops satisfy the conditions g_{i}^{2} = 1 for every
g_{i} ε L_{n}(m). We prove Z_{t}L_{n}(m) has an S-idempotent when t is a perfect number
or when t is of the form 2^{i}p or 3^{i}p (where p is an odd prime) or in general when
t = p_{1}^{i}p_{2} (p_{1} and p_{2} are distinct odd primes). It is important to note that we
are able to prove only the existance of a single S-idempotent; however we leave
it as an open problem wheather such loop rings have more than one S-idempotent.
This paper has three sections. In section one, we give the basic notions about
the loops L_{n}(m) and recall the definition of S-idempotents in rings. In section
two, we establish the existance of S-idempotents in the loop ring Z_{t}L_{n}(m). In
the final section, we suggest some interesting problems based on our study.

**Comments:** 9 pages

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### Submission history

[v1] 11 Mar 2010

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