On τ-injective hulls of modules

Let tau be a hereditary torsion theory on the category R-mod of left modules over an associative unitary ring R. We will study the tau-injective hull of certain modules, giving an explicit description as well. The main goal will be the connection between the tau-injective hull of R/p, for some prime ideal p of a commutative ring. R, and the annihilator of p in the injective hull of R/p. Some torsion theories generalizing the Dickson torsion theory in the case of a commutative ring R will be considered and further properties will be established.