6 PURE-INJECTIVE HULLS OF MODULES OVER VALUATION RINGS

Abstract. If R ̂ is the pure-injective hull of a valuation ring R, it is proved that R ̂ ⊗R M is the pure-injective hull of M, for every finitely generated R-module M. Moreover R ̂ ⊗R M ∼ = ⊕1≤k≤nR̂/AkR̂, 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. The aim of this paper is to study pure-injective hulls of modules over valuation rings. If R is a valuation domain and S a maximal immediate extension of R, then, in [10], Warfield proved that S is a pure-injective hull of R. Moreover, for each finitely generated R-module M, he showed that S ⊗RM is a pure-injective hull of M and a direct sum of gen M indecomposable pure-injective modules. We extend this last result to every valuation ring R by replacing S with the pure-injective hull R ̂ of R. As in the domain case R ̂ is a faithfully flat module. Moreover, for each x ∈ R ̂ there exist r ∈ R and y ∈ 1 + PR ̂ such that x = ry. This property allows us to prove most of the main results of this paper. We extend results obtained b