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

Trace of canonical modules, annihilator of Ext, and classes of rings close to being Gorenstein