Algebraic derivations with constants satisfying a polynomial identity. (Q1425640)

The authors continue the line of investigation started by \textit{I. N. Herstein} and \textit{L. Neumann} [Ann. Mat. Pura Appl., IV. Ser. 102, 37-44 (1975; Zbl 0302.16020)]. The philosophy is that if \(R\) is a unitary algebra over a commutative ring \(C\) and \(b\in R\) is integral over \(C\), then the centralizer \(C_R(b)\) is large enough in the sense that nice properties of \(C_R(b)\) can be usually extended to the whole of \(R\). The paper under review considers a semiprime ring \(R\) with extended centroid \(C\) and a derivation \(\delta\) which is integral over \(C\). The main result is that if the algebra of constants \(R^{(\delta)}\) satisfies a polynomial identity, then \(R\) also satisfies a polynomial identity. This answers affirmatively a problem raised by \textit{M. Smith} [Duke Math. J. 42, 137-149 (1975; Zbl 0336.16020)] and recently again by \textit{J. Bergen} and \textit{P. Grzeszczuk} [J. Algebra 228, No. 2, 710-737 (2000; Zbl 0957.16023)]. (A derivation \(\delta\) of the symmetric Martindale quotient ring \(Q\) of \(R\) is called a continuous derivation of \(R\) if \(\delta(I)\subset R\) for an essential two-sided ideal of \(R\). The authors prove their main result in the more general set-up of continuous integral derivations of \(R\).)