We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature.

中文翻译：

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伯奇理想和伯奇环

我们介绍了伯奇理想和伯奇环的概念。它们很容易定义，并且可以被看作是许多著名概念的概括，例如有限共长的积分闭合理想和最小重数的 Cohen-Macaulay 环。我们给出了这些对象的几个特征。我们表明它们满足许多有趣和理想的特性：理想理论、同调、分类。我们将它们与文献中的其他类别的理想和戒指联系起来。