We recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maximal element in certain Apéry sets.

中文翻译：

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关于单纯仿射半群环的深度

我们回顾并深入研究了仿射半群环深度的不同表征，提供了三维和四维情况下深度二的原始表征，这与某些 Apéry 集合中最大元素的存在密切相关。