This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen-Macaulay local rings that admit a canonical ideal. Here to each such ring with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of a ring. The minimal values of these functions characterize specific classes of Cohen-Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

中文翻译：

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Cohen-Macaulay 环的双正则度

这篇论文是 [8] 的续篇，在那里我们引入了 Cohen-Macaulay 局部环的一个不变式，称为规范度，它承认规范理想。在这里，对于具有典型理想的每个这样的环，我们附加了一个不同的不变量，称为双典型度，它在维度 1 中也出现在 [12] 中作为环的残差。这些函数的最小值表征特定类别的 Cohen-Macaulay 环。我们给出了这些学位的统一表示，并讨论了双规范学位提供的一些计算机会。