Pure-injective hulls of modules over valuation rings

If $\widehat{R} is the pure-injective hull of a valuation ring $R$, it is proved that $\widehat{R}\otimes_RM$ is the pure-injective of $M$, for each finitely generated module $M$. Moreover, $\widehat{R}\otimes_RM\simeq\oplus_{1\leq k\leq n}\widehat{R}/A_k\widehat{R}$, where $A_1,\dots,A_n$ 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.