Recently, Tang et al. [12] (resp. Wu et al. [15]) obtained a necessary and sufficient condition for a finite commutative group ring $\mathbb{F}G$ to be a ⁎-clean ring under the classical involution (resp. the conjugate involution), where $\mathbb{F}$ denotes a finite field and G denotes a finite abelian group. It was shown in [12] (resp. [15]) that $\mathbb{F}G$ is ⁎-clean under the classical involution (resp. the conjugate involution) if and only if the congruence $q^{dx}\equiv -1(\mathrm{mod}m)$ (resp. $q^{2dx}\equiv -q(\mathrm{mod}m)$) has a solution, where q is a prime power relating to the order of the finite field $\mathbb{F}$, 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).

最近，唐等人。[12]（分别是吴等人[15]）获得了有限交换群环的充要条件$\mathbb{F}G$ 是经典对合（分别是共轭对合）下的⁎-清洁环，其中 $\mathbb{F}$表示有限域，G表示有限阿贝尔群。在[12]（第[15]）中显示$\mathbb{F}G$ 当且仅当全等时，在经典对合（分别是共轭对合）下为clean-clean $q^{dX}\equiv -\mathrm{1个}（\mathrm{模}米）$ （分别 $q^{2dX}\equiv -q（\mathrm{模}米）$）有一个解，其中q是与有限域阶次有关的素幂$\mathbb{F}$，d和m是与有限群G有关的正整数。本文继续了这些工作，表明存在一种相当简单的方法来确定这些同余项是否具有解。因此，产生了简单明了的准则来确定给定的有限交换群环在经典对合（分别是共轭对合）下是否为⁎-干净的。