  Over an arbitrary commutative ring, correspondences among three sets, the set
of trace ideals, the set of stable ideals, and the set of birational extensions
of the base ring, are studied. The correspondences are well-behaved, if the
base ring is a Gorenstein ring of dimension one. It is shown that with one
extremal exception, the surjectivity of one of the correspondences
characterizes the Gorenstein property of the base ring, provided it is a
Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring,
a characterization of modules in which every submodule is a trace module is
given. The notion of anti-stable rings is introduced, exploring their basic
properties.
