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Tangential idealizers and differential ideals. (English) Zbl 1342.13039
Let \(k\subset R\) be an extension of commutative Noetherian rings with identity. If \(I\) is an ideal of \(R\) and \(N\) a submodule of an \(R\)-module \(M\), then the tangential idealizer (over \(k\)) of \(I\) with respect to \(N\) is defined as \(T_{R|k}^{M}(I, N)=\{d\in\mathrm{Der}_k(R, M)\mid d(I)\subseteq N\}\). If \(M=R\) and \(N=I\), one writes \(T_{R|k}(I)\) for \(T_{R|k}^{M}(I, N)\).
The paper under review establishes some properties of tangential idealizers; in particular, it is shown that if \(I=\bigcap_{i=1}^tQ_i\) is a primary decomposition of an ideal \(I\) without embedded components, then \(T_{R|k}(I)=\bigcap_{i=1}^{t}T_{R|k}(Q_{i})\). Moreover, if \(I\) is non-differential (i. e., it is not invariant under every \(k\)-derivation of \(R\)), then the last intersection taken over \(\{i\mid Q_i\) is a non-differential\(\}\) is a minimal primary decomposition of \(T_{R|k}(I)\) in \(\mathrm{Der}_k(R, R)\). The author also describes ideals \(I\) whose radicals, as well as ordinary and symbolic powers, have the same tangential idealizer \(T_{R|k}(I)\). The last part of the paper deals with differential ideals (that is, ideals \(I\) such that \(d(I)\subseteq I\) for every \(d\in\mathrm{Der}_k(R, R)\)). Using properties of tangential idealizers, the author gives a characterization of differential ideals and proves that if \(I\) is an ideal of \(R\) having no embedded primary components, then \(I\) is differential if and only if all its primary components are differential.

13N15 Derivations and commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C13 Other special types of modules and ideals in commutative rings
13N99 Differential algebra
Full Text: DOI
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