Radical rings with soluble adjoint groups
An associative ring \(R\) is called radical if \(R\) coincides with its Jacobson radical, that is, when the set of all elements of \(R\) forms a group (denoted by \(R^\circ\) and called the adjoint group of \(R\)) under the operation \(r\circ s=r+s+rs\) on \(R\). If \(r_1,r_2,\dots\) are elements of \(R\), the Lie commutators \([r_1,r_2,\dots,r_{n+1}]\) are defined inductively by \([r_1,r_2]=r_1r_2-r_2r_1\) and \([r_1,r_2,\dots,r_{n+1}]=[[r_1,r_2,\dots,r_n],r_{n+1}]\) for all \(n\geqslant 2\). Every associative ring \(R\) can be considered as a Lie ring under the Lie multiplication \([r_1,r_2]=r_1r_2-r_2r_1\) for all \(r_1,r_2\in R\). For any additive subgroups \(V\) and \(W\) of \(R\), the additive subgroup of \(R\) generated by all Lie commutators \([r_1,r_2]\) with \(r_1\in V\) and \(r_2\in W\) is denoted by \([V,W]\). The derived chain of a Lie ring \(R\) is defined inductively by \(\delta_0(R)=R\) and \(\delta_{n+1}(R)=[\delta_n(R),\delta_n(R)]\) for each integer \(n\geqslant 0\). The ring \(R\) is called Lie-soluble of length at most \(m\) if \(\delta_m(R)=0\). Soluble groups of derived length \(m\) are defined in a corresponding way where the usual group commutator replaces the Lie commutator. A ring \(R\) is called an \(n\)-Engel ring if \([r,s,\dots,s]=0\) for each pair of elements \(r\) and \(s\) of \(R\) where \(s\) appears exactly \(n\)-times. \(m\)-Engel groups are defined correspondingly. An ideal \(I\) of \(R\) is called a commutative ideal if \([I,I]=0\). The authors prove that for a radical ring \(R\), the adjoint group \(R^\circ\) is soluble of derived length at most \(n\) for some positive integer \(n\) if and only if the following two conditions hold: (1) \(R\) is Lie-soluble of length at most \(l\) for some positive integer \(l\) depending only on \(n\), and (2) there exists a chain \(0=I_0\subseteq I_1\subseteq\cdots\subseteq I_m=R\) of ideals of \(R\) such that every factor \(I_i/I_{i-1}\) is generated by commutative ideals of \(R/I_{i-1}\) for \(1\leqslant i\leqslant m\) and the length \(m\) of such a shortest chain of \(R\) depends only on \(n\). The authors also prove that if \(R\) is a radical ring whose adjoint group \(R^\circ\) is soluble of derived length \(n\) for some positive integer \(n\) and \(L\) is the unique maximal locally nilpotent ideal of \(R\), then there exist positive integers \(k,l\) and \(m\) depending only on \(n\) such that the following three conditions hold: (1) \(R\) satisfies the identity \([x,y]^k=0\) for all \(x,y\in R\). (2) The factor ring \(R/L\) is commutative and \(L\) is an \(l\)-Engel ring. (3) The derived subgroup of \(R^\circ\) is an \(m\)-Engel group. An Abelian group \(G\) is said to be of finite torsion-free rank if it has a finitely generated torsion-free subgroup \(A\) such that the factor group \(G/A\) is periodic. The authors prove that if \(R\) is a radical ring whose adjoint group \(R^\circ\) is soluble and the commutator factor group of \(R^\circ\) has finite torsion-free rank, then \(R\) is locally nilpotent. In particular, every radical ring whose adjoint group is finitely generated and soluble must be nilpotent.
scientific article; zbMATH DE number 1740521