The Seriality of the Group Ring of a Finite Group Depends Only on the Characteristic of the Field

Abstract

Let F be a field, and let G be a finite group. In this note, we prove that the seriality of the group ring FG depends only on the characteristic of F . For instance, the ring FpG is serial for a prime field with p elements if and only if the ring FpG is serial, where Fp denotes the algebraic closure of Fp. This solves one problem from the book [3]. We briefly recall basic definitions. A module M is said to be uniserial if its lattice of submodules is a chain; M is serial if it is a direct sum of uniserial modules. The ring R is said to be serial if R is a serial right and serial left R-module. It is well known that the ring R is serial if and only if there exists a complete orthogonal collection of indecomposable idempotents e1, . . . , en ∈ R such that each right module eiR is uniserial, and also each left module Rei is uniserial. Some authors call serial rings Nakayma rings (which is quite justified); therefore serial finite-dimensional algebras over a field are called Nakayama algebras (see [3, Chap. 9]). Let F be an arbitrary field, and let G be a finite group. Of interest to us is the question as to when the group ring FG is serial. Baba and Oshiro suggested the following problem.