Title：On *-clean non-commutative group rings
Abstract: A ring with involution ∗ is called ∗-clean if each of its elements is a sum of a unit and a projection (∗-invariant idempotent). Recently clean rings and ∗-clean rings have been studied intensively by several authors. In this talk, we discuss the group rings of the dihedral groups D2n, and the generalized quaternion groups Q2n with the standard involution ∗. For the non-semisimple group ring case, we characterize the∗-cleanness of RD2pk with a prime p ∈ J(R), and RD2n with 2 ∈ J(R), where R is a commutative local ring and J(R) is its Jacobson radical. For the semisimple group ring case. we investigate when KG is ∗-clean, where K is the field of rational numbers Q or a finite field Fq and G = D2n or G = Q2n. A complete characterization of such group rings is given.
