Pure-injective hulls of modules over valuation rings

If $\hat{R} is the pure-injective hull of a valuation ring$R$, it is proved that$\hat{R}\otimes\_RM$is the pure-injective of$M$, for each finitely generated module$M$. Moreover,$\hat{R}\otimes\_RM\simeq\oplus\_{1\leq k\leq n}\hat{R}/A\_k\hat{R}$, where$A\_1,...,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.