Reversible group rings.
An associative ring \(R\) with identity is said to be reversible if \(ab=0\) implies \(ba=0\) for all \(a,b\in R\). The main problem of this paper is the question when the group ring \(K[G]\) of a group \(G\) over a field \(K\) is reversible. The authors prove that if \(G\) is a non-Abelian torsion group and \(K[G]\) is reversible, then \(G\) is Hamiltonian. Thus, using the structure theorems for Hamiltonian groups and for group rings of finite Abelian groups over fields, they show that the question can be reduced to group rings \(K[Q_8]\), where \(Q_8\) is the quaternion group, and \(p=\text{char}(K)\) must be \(0\) or \(2\). It is proved that if \(p\neq 2\) (resp., \(p=2\)), then \(K[Q_8]\) is reversible if and only if the equation \(x^2+y^2+1=0\) (resp., \(x^2+x+1=0\)) has no solution in \(K\). Let \(G\) be a non-Abelian torsion group and let \(K\) be a field of characteristic \(p\). The main result asserts that \(K[G]\) is reversible if and only if \(G\) is Hamiltonian and \(p=0\) (resp., \(p=2\) and \(G/Q_8\) has no element of even order) and the equation \(x^2+y^2+1=0\) (resp., \(x^2+x+1=0\)) has no solution in any cyclotomic field \(K(\xi_n)\) for any odd \(n\) which is the order of an element of \(G\).
scientific article; zbMATH DE number 2117268