∗-Cleanness of finite group rings

Abstract

ABSTRACTIn this paper, we completely characterize when a group algebra FG of a finite abelian group G over a finite field F is a ∗-clean ring. The main result states that, for a finite abelian group G of order n and a finite field Fq with char(Fq) = p, the group algebra FqG is ∗-clean under the classical involution if and only if there exists a positive integer w such that qdw≡−1(modm), where G(p) is the Sylow p-subgroup of G, m = exp(G∕G(p)) is the exponent of G∕G(p), m2 is the second exponent of G∕G(p) and d is the order of q modulo m2. Particularly, if G = Cn is the cyclic group of order n with gcd(p,n) = 1, then is ∗-clean if and only if there exists a positive integer w such that qw≡−1(modn). Our results improve several known results and give an answer to an existing question in the literature.