A reduction theorem for the existence of *-clean finite group rings

摘要

Recently, Tang et al. [12] (resp. Wu et al. [15]) obtained a necessary and sufficient condition for a finite commutative group ring F G to be a *-clean ring under the classical involution (resp. the conjugate involution), where F denotes a finite field and G denotes a finite abelian group. It was shown in (resp. [15]) that FG is *-clean under the classical involution (resp. the conjugate involution) if and only if the congruence q d(dx ) -1 (mod m) (resp. q (2dx) -q (mod m)) has a solution, where q is a prime power relating to the order of the finite field F, d and m are positive integers relating to the finite group G. This paper continues these works, showing that there is a fairly simple way to determine whether the congruences have solutions. Consequently, explicit and simple criterions are produced to determine whether or not a given finite commutative group ring is *-clean under the classical involution (resp. the conjugate involution). (C) 2020 Elsevier Inc. All rights reserved.