Smarandache Idempotents in Loop Rings [Z.Sub.T][L.Sub.N](M) of the Loops [L.Sub.N](M)‪.‬

Abstract In this paper we establish the existence of S-idempotents in case of loop rings [Z.sub.t][L.sub.n] (m) for a special class of loops [L.sub.n] (m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions [g.sup.2.sub.i] for every [g.sub.i] [member of] [L.sub.n] (m). We prove [Z.sub.t] [L.sub.n] (m) has an S-idempotent when t is a perfect number or when t is of the form [2.sup.i] p or [3.sup.i] p (where p is an odd prime) or in general when t = [p.sup.i]1p2 (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. [section] 1. Basic Results