We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides with symbolic powers in many interesting non-smooth settings, and so it could serve as a generalization of the Zariski–Nagata Theorem.

中文翻译：

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Noetherian运算符，主要子模块和符号能力

我们给出了在Noetherian可交换环的一般类上对于主要子模块的所谓Noetherian算子的存在的代数和独立证明。Noetherian运算符的存在可提供对差分子运算符的主要子模块的等效描述。结果，我们引入了一个新的微分幂概念，它与许多有趣的非光滑环境中的符号幂相符，因此可以作为Zariski-Nagata定理的推广。