# zbMATH — the first resource for mathematics

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.

##### MSC:
 13N05 Modules of differentials 13B10 Morphisms of commutative rings 16W25 Derivations, actions of Lie algebras
Full Text: