Smarandache idempotents in certain types of groups rings

TL;DR

This paper investigates the properties of S-idempotents in group rings over finite cyclic groups, providing conditions under which all nonzero idempotents are S-idempotents and explicitly finding such elements over algebraically closed fields.

Contribution

It establishes criteria for S-idempotents in group rings over inite cyclic groups and characterizes these elements over fields of characteristic zero.

Findings

Conditions for all nonzero idempotents to be S-idempotents in inite cyclic group rings

Explicit identification of S-idempotents over algebraically closed fields of characteristic zero

Characterization of S-idempotents in group rings over inite cyclic groups

Abstract

In this paper we study S-idempotents of the group ring \mathbb{Z}_2G where G is a finite cyclic group of order n. We give a condition on n such that every nonzero idempotent element of the group ring \mathbb{Z}_2G is Smarandache idempotent and we find Smarandache idempotents of the group ring KG, where K is an algebraically closed field of characteristic 0 and G is a finite cyclic group.