Associative rings with metabelian adjoint group.
Every associative ring \(R\) can be viewed as a Lie ring under the Lie multiplication \([r,s]=rs-sr\), for all \(r,s\in R\). For additive subgroups \(V\) and \(W\) of \(R\), the authors denote by \([V,W]\) the additive subgroup of \(R\) generated by all Lie commutators \([v,w]\). The derived chain of a Lie ring \(R\) is defined inductively as \(\delta_0(R)=R\) and \(\delta_{n+1}(R)=[\delta_n(R),\delta_n(R)]\), for each integer \(n\geq 0\). The ring \(R\) is called Lie soluble of length \(m\geq 1\) if \(\delta_m(R)=0\) and \(\delta_{m-1}(R)\neq 0\). Lie soluble rings of length at most 2 are called Lie metabelian. On the other hand, the set of all elements of an associative ring \(R\) forms a monoid under the circle operation \(r\circ s=r+s+rs\) on \(R\). The group of all invertible elements of this monoid is called the adjoint group of \(R\) and is denoted by \(R^\circ\). The ring \(R\) is called radical if \(R=R^\circ\). In this paper the authors give a positive answer to a question of S. Jennings and A. Krasil'nikov by proving that a radical ring \(R\) is Lie metabelian if and only if the adjoint group \(R^\circ\) of \(R\) is metabelian. Moreover, they show that for a ring \(R\) with unity whose multiplicative group \(R^*\) is metabelian, if \(R\) is generated by \(R^*\) and the factor ring \(R/J\) is commutative and Artinian, where \(J\) denotes the Jacobson radical of \(R\), then \(R\) is a Lie metabelian ring. The authors also show that the condition that \(R\) is generated by \(R^*\) cannot be dropped. This enables the authors to conclude that a local ring \(R\) is Lie metabelian if and only if its multiplicative group \(R^*\) is metabelian.
scientific article; zbMATH DE number 2105234