# Quasi-projective modules over integral domains

## Description

We study quasi-projective modules over integral domains. A module M is called quasi-projective if it has the projective property relative to all exact sequences of the form 0 → N → M → M/N → 0, where N is a submodule of M. Quasi-projective modules have been introduced by Miyashita as a generalization of projective modules The main goals of our research are to generalize the results on quasi-projective modules over valuation domains to arbitrary integral domains and to study special types of quasi-projective modules, e.g. uniserial modules The dissertation consists of two parts The first part is concerned with quasi-projective modules over general domains. The main results of the first part are the following. (1) The so-called 1½-generated ideals are quasi-projective, moreover, projective. (2) The quotient field Q of an integral domain R is a quasi-projective R-module if and only if every proper submodule of Q is complete in its R -topology. (3) Integral domains all of whose ideals are quasi-projective are exactly the almost maximal Prufer domains The second part of the dissertation is primarily devoted to quasi-projective uniserial modules over valuation domains. The main results of the second part are the following. (1) Uniserial module U is quasi-projective if and only if it is weakly quasi-projective and an additional technical requirement is satisfied. (2) For torsion free modules of rank 1, quasi-projectivity is equivalent to the weak quasi-projectivity, and the latter is determined by completeness of certain endomorphism rings in their ring topologies. (3) The archimedean ideals of a valuation domain R with non-principal maximal ideal P are quasi-projective if and only if R/K is complete in the R/ K-topology for each archimedean ideal K, not isomorphic to P In conclusion we investigate the influence of quasi-projectivity on the decomposability of modules over valuation domains as well as on the properties of direct sums of such modules. We show that a torsion-free quasi-projective module M over a valuation domain which has a dense basic submodule is completely decomposable and that direct sums of ℵ -generated uniserial modules of cardinality less than ℵ are quasi-projective