High order derivations and primary ideals to regular prime ideals (Q1077470)

Let A be a commutative algebra over a commutative ring k. For any k- module endomorphism f of A and any element x of A, let [f,x] denote the endomorphism defined by the rule: \([f,x](y)=f(xy)-x\cdot f(y)-y\cdot f(x).\) The endomorphism f is a derivation if, and only if, \([f,x]=0\) for all x in A. A derivation is called a derivation of order 1, and f is called a derivation of \(order\quad n\) if [f,x] is a derivation of order \(n-1\) for all x in A. Let Der\(^ n_ k(A)\) denote the k-module of all derivations of order n, and note that Der\(_ k^{n-1}(A)\subseteq\) Der\(_ k^ n(A)\). A submodule U of Der\(^ n_ k(A)\) is called admissible if \([f,x]\in U\) whenever \(f\in U\) and \(x\in A\). Letting p be any prime ideal of A, the author establishes a Galois connection between the lattice of p-primary ideals of exponent \(n+1\) in A and the lattice of admissible submodules of Der\(^ n_ k(A)\). He then looks for the Galois closed p-primary ideals of exponent \(n+1.\) The main result is: If k is an integral domain and A is a polynomial or power series ring in indeterminates \(X_ 1,...,X_ m\), then the primary ideals belonging to \(p=(X_ 1,...,X_ m)\) and having finite exponent are Galois closed. Additional results are obtained for an algebra A over a perfect field and a prime ideal p such that the localization \(A_ p\) is a regular local ring. Examples show that there may exist admissible submodules of Der\(^ n_ k(A)\) which are not Galois closed, and when \(A_ p\) is not regular there may exist p-primary ideals of finite exponent which are not Galois closed.