Left ideals and derivations in semiprime rings.
Let \(R\) be a prime or semiprime ring with center \(Z(R)\), let \(L\) be a nonzero left ideal, and let \(D\) and \(E\) be nonzero derivations on \(R\). For \(S\subseteq R\), denote by \((S)\) the ideal generated by \(S\). The author explores the following conditions, suggested by a classic paper of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)]: (i) \(DE\) acts like a derivation on \(L\); (ii) \([D(x),x]\in Z(R)\) for all \(x\in L\); (iii) \(xD(x)\in Z(R)\) for all \(x\in L\); (iv) \(D(x)x-xE(x)\in Z(R)\) for all \(x\in L\). Among other results are the following: (A) If \(R\) is semiprime and 2-torsion-free and satisfies (i), then \(LD(L)\subseteq\text{ann}((E(R)))\) and \(LE(L)\subseteq\text{ann}((D(R)))\); (B) If \(R\) is semiprime and satisfies (iv), then \((D(L))\), \((D(R)L)\), \((LD(R))\) and \((LE(L))\) are all central. This paper is the latest in a long sequence of papers by various authors, all motivated by Posner's theorems; and the author includes a helpful description of earlier results and an extensive bibliography.
scientific article; zbMATH DE number 2105244
scientific article