Algebra

Smarandache Idempotents in Loop Rings Z_tL_n(m) of the Loops L_n(m):

Authors: W.B.Vasantha, Moon K. Chetry
Comments: 9 pages

In this paper we establish the existance of S-idempotents in case of loop rings Z_tL_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_i² = 1 for every g_i ε L_n(m). We prove Z_tL_n(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 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_tL_n(m). In the final section, we suggest some interesting problems based on our study.
Category: Algebra