Lie ideals with regular and nilpotent elements and a result on derivations
The results in this paper determine the structure of semi-prime rings if either the image of a derivation, or a Lie ideal, has each element satisfying certain conditions. For example, if R is a ring with no nonzero nil left ideal and if d is a derivation of R with every value, d(r), either invertible or nilpotent, then either \(d=0\), R is a division ring, or \(R=M_ 2(D)\) for D a division ring. The same conclusion holds if one assumes only that R is a semi-prime ring but insists that the indices of nilpotence of the d(r) are bounded. To prove these theorems, the author makes use of the known results when all values d(r) are nilpotent [\textit{B. Felzenszwalb} and \textit{C. Lanski}, J. Algebra 83, 520- 530 (1983; Zbl 0519.16022); and \textit{A. Giambruno} and \textit{I. N. Herstein}, Rend. Circ. Mat. Palermo, II. Ser. 30, 199-206 (1981; Zbl 0482.16030)] in order to reduce to the case of a primitive ring, at which point a clever computation using the Jacobson density theorem finishes the proof. Similar theorems are obtained for Lie ideals of R. Specifically, if U is a Lie ideal of R having each of its elements nilpotent or regular, then if R has no nonzero nil left ideal, either U is central, R is a domain, or R is an order in \(M_ 2(F)\) for F a field. The same conclusion holds if one bounds the indices of nilpotence of the elements of U and assumes only that R is a semi-prime ring. The proofs of these results make use of the theorems above for derivations.
scientific article; zbMATH DE number 3855335
scientific article