In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and show that they are almost Gorenstein rings in dimension one, and are closely related to Teter rings in the artinian case. We explain their basic properties and give some examples.

中文翻译：

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规范模块的踪迹、Ext 的湮灭器和接近 Gorenstein 的环类

在本笔记中，我们研究规范模块的迹理想。给出了在某些 Ext 模块的歼灭器方面的迹理想的特征。我们应用我们的结果来研究最近文献中出现的许多类接近 Gorenstein 的环。我们讨论了由 Herzog-Hibi-Stamate 引入的接近 Gorensteinness 的行为，在规则序列的减少下。提出了立川猜想的近似 Gorenstein 版本，并对类型 2 环给出了肯定的答案。我们还使用规范模引入了弱近乎 Gorenstein 环的类，并表明它们在一维上几乎是 Gorenstein 环，并且与阿提恩情况下的 Teter 环密切相关。我们解释了它们的基本属性并给出了一些例子。