Pure-injective hulls of modules over valuation rings

If image is the pure-injective hull of a valuation ring R, it is proved that image is the pure-injective hull of M, for every finitely generated R-module M. Moreover image, where (Ak)1≤k≤n is the annihilator sequence of M. The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module are isomorphic.