Semilocal rings whose adjoint group is locally supersoluble.
Let \(R\) be an associative ring, not necessarily with an identity element. Let \(R^{ad}\) be the adjoint semigroup of \(R\) under the operation \(a\circ b=a+b+ab\) for all \(a,b\in R\) with neutral element \(0\in R\). Let \(R^\circ\) be the adjoint group of \(R\), that is, the group of all invertible elements of the semigroup \(R^{ad}\). The Jacobson radical of \(R\) is its unique ideal \(J=\text{Jac}(R)\) which is maximal with respect to the condition that \(J\subseteq R^\circ\). A ring \(R\) is called radical if \(R=R^\circ\) which means that \(R=\text{Jac}(R)\). The ring \(R\) is Artinian if it satisfies the minimum condition for one-sided ideals and semilocal if the factor ring \(R/\text{Jac}(R)\) is Artinian. Every ring \(R\) can be considered as a Lie ring under the Lie multiplication \([r,s]=rs-sr\) for all \(r,s\in R\). An additive subgroup \(L\) of \(R\) is a Lie-ideal if \([L,R]\subseteq L\), where \([L,R]\) is the additive subgroup of \(R\) generated by all commutators \([x,r]\) with \(x\in L\) and \(r\in R\). The ring \(R\) is Lie-nilpotent if it has a finite series \(0=L_0\subseteq L_1\subseteq\cdots L_n=R\) of Lie-ideals such that \([L_i,R]\subseteq L_{i-1}\) for all \(i\geqslant 1\). In this case \(R\) has also an ascending series of Lie-ideals whose factors are cyclic as additive groups. Each ring \(R\) with this latter property is called Lie-supersoluble. A group with an ascending series of normal subgroups whose factors are cyclic is called supersoluble. A ring \(R\) is locally Lie-nilpotent or locally Lie-supersoluble if every finitely generated subring of \(R\) is Lie-nilpotent or Lie-supersoluble, respectively. In this paper the authors show that if \(R\) is an associative ring and \(R\) is radical, then the adjoint group \(R^\circ\) is locally supersoluble if and only if \(R^\circ\) is locally nilpotent and so \(R\) is locally Lie-nilpotent. Moreover, if \(R\) is semilocal and its adjoint group \(R^\circ\) is locally supersoluble, then \(R\) is locally Lie-supersoluble and contains a locally Lie-nilpotent ideal \(I\) of finite index such that the factor ring \(R/I\) is a direct sum of ideals each of which is isomorphic either to the Galois field \(\mathbb F_p\) of a prime order \(p\) or to the full matrix ring \(M_2(\mathbb F_2)\) over \(\mathbb F_2\).
scientific article; zbMATH DE number 5797314