Identities with Engel conditions on derivations.
Let \(R\) be a semiprime ring with center \(Z(R)\), symmetric Martindale quotient ring \(Q\), and nonzero derivation \(D\). The authors prove two results involving \(D\) and Engel conditions. Let \([x,y_1]_1=xy_1-y_1x\) and for \(n>1\) set \([x,y_1,\dots,y_n]_n=[[x,y_1,\dots,y_{n-1}]_{n-1},y_n]_1\). The first major result of the authors is that for fixed positive integers \(s\), \(m\), \(n_1,\dots,n_s\), if the expression \([x^m,D(x^{n_1}),\dots,D(x^{n_s})]_s=0\) for all \(x\in R\), and if \(R\) is \(k!\)-torsion free for \(k=1+\max\{m,n_j\}\), then \(D(R)\subseteq Z(R)\). The second main result replaces each \(D(x^{n_j})\) above with \(D(x)^{n_j}\) and concludes, without the \(k!\)-torsion free condition, that there is a central idempotent \(e\in Q\) so that the extension of \(D\) to \(Q\) is zero on \(eQ\) and that \((1-e)Q\) satisfies the standard polynomial identity \(S_4\). Further, for this second result, when \(R\) is prime, then it must satisfy \(S_4\) and if all the \(n_j\) are odd, then \(\text{char\,}R=2\).
scientific article; zbMATH DE number 6018281
scientific article