Smarandache Idempotents in Loop Rings

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 g 2 i = 1 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 = pi1p2 (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. § 1 : Basic Results Here we just give the basic notions about the loops Ln(m) and the definition of Sidempotents in rings. Definition 1.1 [4]: Let R be a ring. An element x ∈ R \ {0} is said to be a Smarandache idempotent (S-idempotent) of R if x = x and there exist a ∈ R \ {x, 0} such that i. a = x ii. xa = x or ax = a. For more about S-idempotent please refer [4]. Definition 1.2 [2]: A positive integer n is said to be a perfect number if n is equal to the sum of all its positive divisors, excluding n itself. e.g. 6 is a perfect number. As