On quasi-projective uniserial modules (Q1408586)

The author studies quasi-projective uniserial modules over a valuation domain \(R\) and, in particular, quasi-projective ideals of \(R\). The quasi-projectivity of a uniserial \(R\)-module \(U\) is characterized in terms of lifting of endomorphisms of factors of \(U\) and this characterization is used to describe quasi-projective ideals of \(R\) in terms of the completeness of certain localizations of factor rings of \(R\). In the case of archimedean ideals this description becomes more explicit and it is shown that, if the maximal ideal \(P\) of \(R\) is infinitely generated, a non-principal archimedean ideal \(I\) is quasi-projective if and only if \(R/K\) is complete in the \(R/K\)-topology for each archimedean ideal \(K\not\cong P\) (and, in this case, all archimedean ideals of \(R\) are quasi-projective).