Smarandache Idempotents in Loop Rings Z t L n ( m ) of the Loops L n ( m )

In this paper we establish the existence 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 g i for every gi ∈ Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number or when t is of the form 2p or 3p (where p is an odd prime) or in general when t = p1p2(p1 and p2 are distinct odd primes), It is important to note that we are able to prove only the existence of a single S-idempotent; however we leave it as an open problem whether such loop rings have more than one S-idempotent.