  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.
