Semilocal rings with \(n\)-Engel multiplicative group.
Let \(R\) be an associative ring with a unit element \(1\). \(R\) is called local if the factor ring \(R/J\) is a division ring and \(R\) is semilocal if \(R/J\) is an Artinian ring. Every ring \(R\) may be considered as a Lie ring under the Lie multiplication \([r_1,r_2]=r_1r_2-r_2r_1\) for all \(r_1,r_2\in R\). 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 \([r_1,r_2,\dots,r_{n+1}]=[[r_1,r_2,\dots,r_n],r_{n+1}]\) for all \(n\geq 2\). The ring \(R\) is called Lie nilpotent of class \(n\), if \([r_1,r_2,\dots,r_{n+1}]=0\) for all elements \(r_1,r_2,\dots,r_{n+1}\) of \(R\) and \(n\) is the least integer with this property. Moreover, \(R\) is Engel if \([r,s,\dots,s]=0\) for each pair of elements \(r\) and \(s\) of \(R\), and \(n\)-Engel if \(s\) appears exactly \(n\)-times. Also, \(R\) is a locally Lie nilpotent ring if every finitely generated subring of \(R\) is Lie nilpotent. Nilpotent, Engel and \(n\)-Engel groups are defined in a similar way where the usual group commutator replaces the Lie-commutator. In this paper the authors prove two major results: Theorem A. The multiplicative group \(R^*\) of a local ring \(R\) is \(m\)-Engel for some positive integer \(m\) if and only if \(R\) is \(n\)-Engel as a Lie ring for some positive integer \(n\) depending only on \(m\). Theorem B. Let \(R\) be a ring with \(1\) whose multiplicative group \(R^*\) is \(m\)-Engel for some integer \(m\geq 1\) and let \(J\) be the Jacobson radical of \(R\). If the factor ring \(R/J\) is commutative and \(R\) is generated by \(R^*\) as a ring, then \(R\) is an \(n\)-Engel ring for some integer \(n\geq 1\) depending only on \(m\). As a consequence of Theorem B the authors obtain the following Corollary. Let \(R\) be a semilocal ring whose multiplicative group \(R^*\) is \(m\)-Engel for some integer \(m\geq 1\). Then \(R^*\) is a locally nilpotent group and, furthermore, \(R\) is \(n\)-Engel as a Lie ring, provided that \(R\) is generated by \(R^*\). Also, the authors observe that in Theorem B and its corollary the condition that \(R\) is generated by \(R^*\) cannot be dropped.
scientific article; zbMATH DE number 2138529