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A reduction theorem for the existence of ⁎-clean finite group rings
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2020-04-01 , DOI: 10.1016/j.ffa.2020.101674
Bocong Chen , Jing Huang

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 qdx1(modm) (resp. q2dxq(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).



中文翻译:

clean-清洁有限群环的存在性的约化定理

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FG 是经典对合(分别是共轭对合)下的⁎-清洁环,其中 F表示有限域,G表示有限阿贝尔群。在[12](第[15])中显示FG 当且仅当全等时,在经典对合(分别是共轭对合)下为clean-clean qdX-1个 (分别 q2dX-q)有一个解,其中q是与有限域阶次有关的素幂Fdm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。

更新日期:2020-04-01
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