The Number of Idempotents in Abelian Group Rings

Peter V. Danchev

Suppose that $R$ is a commutative unitary ring of arbitrary characteristic and $G$ is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set $id(RG)$, consisting of all idempotent elements in the group ring $RG$. It is explicitly calculated only in terms associated with $R$, $G$ and their divisions. This result strengthens previous estimates obtained in the literature recently.