On *-Clean Group Rings II

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C₃, C₄, S₃, and Q₈. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebra FC_p is *-clean, where F is a field and C_p is a cyclic group of prime order p. In this article, we extend the above mentioned result from FC_p to F_qC_p^k, where F_q is a finite field and C_p^k is a cyclic group of an odd prime power order p^k. For the general case when G = C_n is cyclic group of order n, we also provide a necessary condition and a few sufficient conditions for F_qC_n to be *-clean.