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

Abstract

Recently, Tang et al. [12] (resp. Wu et al. [15]) obtained a necessary and sufficient condition for a finite commutative group ring FG 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 [12] (resp. [15]) that FG is ⁎-clean under the classical involution (resp. the conjugate involution) if and only if the congruence qdx≡−1(modm) (resp. q2dx≡−q(modm)) 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).