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
[v1] 11 Mar 2010
Unique-IP document downloads: 97 times
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.