A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space ExtnR(R/m,M) vanishes if n ≠ height(m) and is 1-dimensional if n = height(m).
In particular if the ring is local the dualizing module is unique up to isomorphism.
A Noetherian ring does not necessarily have a dualizing module.
In particular any complete local Cohen–Macaulay ring has a dualizing module.
The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R. The ring Z[√–5] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.