Kurosch invariants for torsion-free modules over Nagata valuation domains
For an integral domain \(R\), \(\text{fr}(R)=n\) means that there exists an indecomposable torsion-free \(R\)-module of rank \(n\) and every torsion-free \(R\)-module of finite rank greater than \(n\) is decomposable. In the present paper the author investigates \(\text{fr}(R)\) for a non-complete discrete rank one valuation domain \(R\) (i.e. Nagata valuation domain) and direct decompositions of torsion-free \(R\)-modules. Let \(R^*\) be the completion of \(R\) and let \(Q\), \(Q^*\) be the respective fields of quotients. The author examines the case when \([Q^*:Q]=2\) or 3. In the case of degree 2 he proves a final result that \(\text{fr}(R)=2\) (theorem 8). For the proof he uses Kurosch matrix invariants of torsion-free \(R\)-modules and some relationships between decomposition of torsion-free \(R\)-module and decomposition of a representative matrix. In the case of \([Q^*:Q]=3\) he obtains several results on direct decompositions of torsion-free modules (for proofs he uses also Kurosch invariants). For example, if \(R\) is a Nagata valuation domain with \([Q^*:Q]=3\) and \(A\) is a torsion-free \(R\)-module of rank 4 then \(A\) is decomposable (theorem 15).
direct decompositions of torsion-free modules
scientific article; zbMATH DE number 166143