Pure-injective hulls of modules over valuation rings

If R̂ is the pure-injective hull of a valuation ring RR, it is proved that R̂⊗RM is the pure-injective hull of MM, for every finitely generated RR-module MM. Moreover R̂⊗RM≅⊕1≤k≤nR̂/AkR̂, where (Ak)1≤k≤n(Ak)1≤k≤n is the annihilator sequence of MM. 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.