On \(\ast\)-clean non-commutative group rings (Q2816919)

If every element of an associative ring \(R\) with a unit is a sum of a unit and an idempotent, then \(R\) is called clean. If there exists an operation \(\ast:R\rightarrow R\), such that NEWLINE\[NEWLINE(x + y)^{\ast} = x^{\ast}+ y^{\ast}, \quad (xy)^{\ast} = y^{\ast}x^{\ast} \quad\text{and}\quad (x^{\ast})^{\ast}=xNEWLINE\]NEWLINE for all \(x, y \in R\), then the ring R is called a \(\ast\)-ring or a ring with an involution \(\ast\). If \(p\) belongs to a \(\ast\)-ring \(R\) and \(p^{\ast} = p = p^2\), then \(p\) is called a projection. If each element of \(R\) is a sum of a unit and a projection, then the ring \(R\) is called \(\ast\)-clean. \textit{L. Vaš} [J. Algebra 324, No. 12, 3388--3400 (2010; Zbl 1246.16031)] and also \textit{C. Li} and \textit{Y. Zhou} and investigated the \(\ast\)-clean rings [J. Algebra Appl. 10, No. 6, 1363--1370 (2011; Zbl 1248.16030)]. \textit{Y. Gao} et al. gave an answer of the question when the group rings \(RG\) are \(\ast\)-clean for a commutative local ring \(R\) and for a group \(G\) of the type \(C_3\), \(C_4\), \(S_3\) and \(Q_8\), where \(C_n\) is the cyclic group of order \(n\), \(S_3\) is the symmetric group of degree 3, and \(Q_8\) is the quaternion group of order 8 [Algebra Colloq. 22, No. 1, 169--180 (2015; Zbl 1316.16018)].NEWLINENEWLINE\textit{Y. Li} et al. gave necessary and sufficient conditions for \(FC_p\) to be \(\ast\)-clean in the terms of irreducible factorizations of a \(p\)th cyclotomic polynomial \(\Phi_p(x)\), where \(F\) is a field and \(p\) is a prime number [J. Algebra Appl. 14, No. 1, Article ID 1550004, 11 p. (2015; Zbl 1318.16024)]. They determined also when the group ring \(RC_n\) is \(\ast\)-clean, where \(R\) is a commutative local ring and \(3\leq n \leq 6\). The investigation for \(FC_p\) is extended to the group algebra \(\mathbb F_qC_{p^{k}}\) for a finite field \(\mathbb F_q\) of order \(q\) and a cyclic group \(C_{p^{k}}\) from \textit{H. Huang} et al. [Commun. Algebra 44, No. 7, 3171--3181 (2016; Zbl 1355.16019)]. They gave few sufficient conditions and also a necessary condition for \(F_q C_{n}\) to be \(\ast\)-clean. Suppose that \(R\) is a commutative local ring, \(J(R)\) is the Jacobson radical of \(R\) and \(D_{2n}\) is the dihedral group of order \(2n\). In Section 3, the authors prove that (i) If \(p\) is a prime and \(p\in J(R)\), then the group ring \(RD_{2^{p^k}}\) is \(\ast\)-clean (Theorem 3.1)and (ii) If \(2\in J(R)\) and \(n\) is not a power of 2, then the group ring \(RD_{2n}\) is not \(\ast\)-clean (Theorem 3.2). Suppose \(F_q\) is a finite field of order \(q\), \(\mathbb Q\) is the field of the rational numbers and \(Q_{2n}\) is the quaternion group of orders \(2n\). In Section 4, the authors (i) prove that the group rings \(\mathbb QD_{2n}\) and \(F_qD_{2n}\) with \(\mathrm{gcd}(q,2n) =1\) are \(\ast\)-clean and (ii) give necessary and sufficient conditions the group ring \(\mathbb Q Q_{2n}\) to be \(\ast\)-clean. They provide also (i) sufficient conditions the group ring \(\mathbb F_qQ_{2n}\) to be \(\ast\)-clean and (ii) construct a set of primitive central idempotents and the Wedderburn decompositions of the group rings \(\mathbb F_qD_{2n}\), \(\mathbb F_qQ_{2n}\) and \(\mathbb QQ_{2n}\). In Section 5, the authors present two examples in which the group rings are \(\ast\)-clean. \textit{Remark.} Theorems 3.1 and 3.2 imply immediately the following necessary and sufficient condition, which the authors not mentioned. \textit{Corollary}. Let \(R\) be a commutative local ring and \(2\in J(R)\). The group ring \(RD_{2n}\) is \(\ast\)-clean if and only if \(n=2^k\), where \(k\) is a natural number.