On *-clean group rings

Abstract

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and 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. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.