The Units and Idempotents in The Group Ring $K(\mathbb{Z}_m \times \mathbb{Z}_n)$

Commun. Korean Math. Soc. 2000 Vol. 15, No. 4, 597-603

Won-Sun Park Chonnam National University

Abstract : Let $K$ be an algebraically closed field of characteristic 0 and let $G = \Bbb Z_m \times \Bbb Z_n$. We find the conditions under which the elements of the group ring $KG$ are units and idempotents respectively by using the represented matrix. We can see that if $\alpha = \sum r(g)g \in KG $ is an idempotent then $r(1)= 0, \frac{1}{mn},\frac{2}{mn}, \cdots, \frac{mn-1}{mn}\; \text{or}\;1. $