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Derivations satisfying polynomial identities. (English) Zbl 0702.13017
Let \(R\) be a commutative ring with identity and \(D\) a family of derivations of \(R\). A \(D\)-module is a left \(R\)-module \(M\) together with a specified mapping \(\bar{}: D\to \mathrm{Hom}_ Z(M,M)\) with \(\bar d(rm)=r\bar d(m)+d(r)m\) for \(d\in D\), \(r\in R\), \(m\in M\). The ring of differential polynomials over \(D\) with coefficients in \(R\) is denoted by \(R[D]\).
The purpose of the paper is to study ideals \(\mathrm{Ann}_{R[D]}M\) of the ring \(R[D]\). A preparatory result is: If \(A\) is an ideal in \(R[D]\) and \(f\) an element in \(A\) such that \(\deg(f)=n\geq 1\) and \(f=\sum^{k}_{i=1}a_ it_{i1}\dots t_{in}+g \), where \(t_{i1},\dots,t_{in}\) for \(i=1,\dots,k\) are distinct monic monomials of degree \(n,\) and \(\deg(g)\leq n\), \(a_ 1,\dots,a_ k\in R\). Then, for any \(r_ 1,\dots,r_ n\in R\), the element \(\sum_{\sigma} \sum^{k}_{i=1} a_ id_{i\sigma(1)}(r_ 1)\dots d_{i\sigma(n)}(r_ n) \), where \(\sigma\) runs over all permutations of the set \(\{1,\dots,n\}\), belongs to \(A\cap R.\)
The main results of the paper are:
If \(A\) is a \(D\)-ideal in \(R\), then \(\mathrm{Ann}_{R[D]}(R[D]A)=R[D]\;\mathrm{Ann}_{R[D]}A.\)
If \(\mathrm{Ann}_{R[D]}R=0\) and \(\mathrm{Ann}_ RM=0\), then \(\mathrm{Ann}_{R[d]}M=0.\)
Further, the constant term of polynomials in \(\mathrm{Ann}_{R[d]}A\) is studied and some properties of \(\mathrm{Ann}_{R[D]}M\) in the case of one derivation i.e. \(D=\{d\}\) are deduced.

13N05 Modules of differentials
13B10 Morphisms of commutative rings
16W25 Derivations, actions of Lie algebras
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