Example 1

If the ring $A$ is local and artinian, the dualizing $A$-modules are the finitely generated injective $A$-modules $\Omega$ such that $\mathrm{Hom}_A(\kappa_A, \Omega)$ is of dimension 1 ($\S$ 3, No. 3, prop. 6), that is to say the Matlis $A$-modules ($\S$ 8, No. 3).