The Krull-Remak-Schmidt decomposition of commutative group algebras

Abstract

We provide the Krull-Remak-Schmidt decomposition of group algebras of the form $k[G]$ where $k$ is a field, which includes fields with prime characteristic, and $G$ a finite abelian group. We achieved this by studying the geometric equivalence of $k[G]$ which we call circulant coordinate rings.