## 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

**Download:** **PDF**

### Submission history

[v1] 11 Mar 2010

**Unique-IP document downloads:** 88 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*