LetRbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of rightR-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image ofRcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable rightR-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime idealPthe annhilator of a finitely generatedP-tame rightR-module cannot be a prime ideal properly contained inP. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.