Goldie conditions for constants of algebraic derivations of semiprime algebras (Q690074)

The authors prove results relating chain conditions in a ring to those of the subring of constants of an algebraic derivation. All chain conditions mentioned are for right ideals. For a ring \(R\) with derivation \(d\), let \(\dim R\) be the Goldie rank of \(R\), \(R^ d = \text{Ker }d\), and \(S = \{x \in R \mid d^ i(x) = 0\text{ for some }i \geq 1\}\). When \(R\) is an algebra over the field \(F\), \(d\) is called algebraic if it is algebraic as an \(F\)-linear transformation on \(R_ F\). The first results assume that \(d\) is nilpotent: that is, \(R = S = \{x \in R\mid d^ n(x) = 0\}\) with \(n\) fixed. In this case, if \(S\) is Artinian, Noetherian, or has \(\dim S\) finite, then the same is true of \(R\). When \(R\) is semi-prime, then \(\dim R\) finite forces \(\dim S\) finite. Using this, the authors consider a general algebraic derivation and show that when \(R\) is semi-prime, if any of \(\dim R\), \(\dim R^ d\), or \(\dim S\) is finite, so are the other two, and that \(R\) is Artinian if and only if \(S\) is. Thus \(R^ d\) Artinian implies that \(R\) is Artinian also. The last main result of the paper is the following: Theorem. Let \(R\) and \(R^ d\) be semi-prime, and let \(d\) be algebraic. Then \(R\) is Goldie (or Artinian) exactly when \(R^ d\) is Goldie (or Artinian). When \(R\) and \(R^ d\) are Goldie with quotient rings \(Q(R)\) and \(Q(R^ d)\), then \(Q(R) = R_ T\), the localization of \(R\) at the set \(T\) of regular elements of \(R^ d\), and \(Q(R)^ d = Q(R^ d)\), where \(d\) naturally extends to \(Q(R)\). An application of this theorem has the same conclusions when \(R\) has no nonzero nilpotent elements and ``\(d\)'' is replaced with \(L\), a finite dimensional solvable Lie algebra acting on \(R\) by algebraic derivations.