Pure-injective hulls of modules over valuation rings

If R(overbar) is the pure-injective hull of a valuation ring R, it is proved that R(overbar)circle times(R) M is the pure-injective hull of M, for every finitely generated R-module M. Moreover R(overbar)circle times(R) M congruent to circle plus(1) <=(k)<=(n) R/A(k) R, where (A(k))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. (c) 2005 Elsevier B.V. All rights reserved.