  Let R be a commutative noetherian local ring. A finitely generated R-module C
is semidualizing if it is self-orthogonal and satisfies the condition
Hom_R(C,C) \cong R. We prove that a Cohen-Macaulay ring R with dualizing module
D admits a semidualizing module C satisfying R\ncong C \ncong D if and only if
it is a homomorphic image of a Gorenstein ring in which the defining ideal
decomposes in a cohomologically independent way. This expands on a well-known
result of Foxby, Reiten and Sharp saying that R admits a dualizing module if
and only if R is Cohen--Macaulay and a homomorphic image of a local Gorenstein
ring.
