Rank-two torsion-free modules over valuation domains
This paper represents the first step in developing the theory of torsion free R-modules of finite rank for general valuation rings R symmetrically to the theory of finitely generated torsion R-modules. This symmetry is known for some special cases of R (e.g. for maximal valuation domains), on the other hand this symmetry is lost if R is almost maximal which fails to be maximal. The principal role in the paper have the class \({\mathcal F}_ 2(R)\) of indecomposable torsion-free R-modules of rank two and the class \({\mathcal G}_ 2(R)\) of two-generated indecomposable torsion R-modules. The authors present a universal construction of indecomposable torsion-free R-modules M of rank two (R non almost maximal) using a unit u of a maximal immediate extension S of R theorem 3.1). This construction is connected with compatible triples (L,H,I) of submodules of the quotient field of R, where L is isomorphic to a basic submodule B of M, H is isomorphic to M/B and \(I=\{a\in R: u\not\in aR+S\}\) is the so-called breath ideal of u. The authors receive then theorem 5.3, which characterizes isomorphic elements of \({\mathcal F}_ 2(R)\) and subsequently they receive a complete and independent set of invariants for \({\mathcal F}_ 2(R)\) in a form of some equivalence classes \([L,H,u]\). Then they show that symmetrical results hold for \({\mathcal G}_ 2(R)\), too.
rank-two torsion-free modules over valuation domains
scientific article; zbMATH DE number 4122096
scientific article