Torsion-free covers and pure-injective envelopes over valuation domains
The purpose of this paper is to describe some relations among the torsionfree covers of cyclic modules, the pure-injective hulls of ideals, the maximal immediate extensions of localizations and the injective hulls of cyclics when the ring R under consideration is a valuation domain. The pure-injective hull of an ideal I of R is described as the ideal generated by I in a maximal immediate extension of the localization of R at a suitable prime ideal determined by I. The pure-injective hull of I and the injective hull E(R/I) of the cyclic module R/I are found to be correspondent in Matlis' equivalence between torsionfree cotorsion modules and torsion h-divisible modules. The endomorphism ring of the pure-injective hull of I is found to be exactly the endomorphism ring of E(R/I), or, alternatively, the R-endomorphism ring of any maximal immediate extension of the localization of R at the prime ideal determined by I. A description is given of the torsionfree covers of the uniserial modules that are homomorphic images of R-submodules of the field of fractions of R. Finally various characterizations are given of valuation domains one of whose localization is maximal or almost maximal making use of the pure-injectivity of ideals, the torsionfree covers and the vanishing of homological functors.
scientific article; zbMATH DE number 3943995
scientific article