Pure-injective hulls of modules over valuation rings

If R̂ is the pure-injective hull of a valuation ring R, it is proved that R̂ ⊗R M is the pure-injective hull of M , for every finitely generated Rmodule M . Moreover R̂ ⊗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 ⊗R M 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 R̂ of R. As in the domain case R̂ is a faithfully flat module. Moreover, for each x ∈ R̂ there exist r ∈ R and y ∈ 1 + PR̂ such that x = ry. This property allows us to prove most of the main results of this paper. We extend results obtained by Fuchs and Salce on pure-injective hulls of uniserial modules over valuation domains ([5, chapter XIII, section 5]). We show that the length of any pure-composition series of a polyserial module M is its Malcev rank Mr M and its pure-injective hull M̂ is a direct sum of p indecomposable pure-injective modules, where p ≤ Mr M . But it is possible to have p < Mr M and we prove that the equality holds for all M if and only if R is maximal (Theorem 4.5). This result is a consequence of the fact that R is maximal if and only if R/N and RN are maximal, where N is the nilradical of R (Theorem 4.4). If U1, . . . , Un are the factors of a pure-composition series of a polyserial module M then the collection (R̂ ⊗R Uk)1≤k≤n is uniquely determined by M . To prove this, we use the fact that R̂ ⊗R U is an unshrinkable uniserial T -module for each uniserial R-module U , where T = EndR(R̂). When R satisfies a countable condition, the collection of uniserial factors of a polyserial module M is uniquely determined by M (Proposition 3.7). In this paper all rings are associative and commutative with unity and all modules are unital. As in [3] we say that an R-module E is divisible if, for every r ∈ R and x ∈ E, (0 : r) ⊆ (0 : x) implies that x ∈ rE, and that E is fp-injective(or absolutely pure) if ExtR(F,E) = 0, for every finitely presented R-module F. A ring R is called self fp-injective if it is fp-injective as R-module. An exact sequence 0 → F → E → G → 0 is pure if it remains exact when tensoring it with any R-module. In this case we say that F is a pure submodule of E. Recall that a module E is fp-injective if and only if it is a pure submodule of every overmodule. A module is said to be uniserial if its submodules are linearly ordered