Smarandache Idempotents in Loop Rings ZtLn(m) of the Loops Ln(m):

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

In this paper we establish the existance of S-idempotents in case of loop rings ZtLn(m) for a special class of loops Ln(m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions gi2 = 1 for every gi ε Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number or when t is of the form 2ip or 3ip (where p is an odd prime) or in general when t = p1ip2 (p1 and p2 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 Ln(m) and recall the definition of S-idempotents in rings. In section two, we establish the existance of S-idempotents in the loop ring ZtLn(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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