Warfield duality and rank-one quasi-summands of tensor products of finite rank locally free modules over Dedekind domains. (English) Zbl 0682.13001
If G and H are strongly indecomposable finite rank torsion free modules over a Dedekind domain and if A is a locally free rank-one quasi-summand of \(G\otimes H\), then G and H are also locally free t(A)-bounded with non-trivial traces and H is quasi-isomorphic to Hom(G,A). The adjointness isomorphism Hom(G\(\otimes H,A)\cong Hom(G,Hom(H,A))\) takes split quasi- epimorphisms to split quasi-monomorphisms. Some results of Warfield from 1968 are looked over in a new light.
Reviewer: R.M.Dimitrić
MSC:
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 13C13 | Other special types of modules and ideals in commutative rings |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
References:
| [1] | Arnold, D. M., Finite Rank Torsion Free Abelian Groups and Rings, (Lecture Notes in Mathematics, Vol. 931 (1982), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0329.20033 |
| [2] | Bourbaki, N., Algèbre, (Modules et Anneaux Semi-Simple (1958), Herman: Herman Paris), Chap. 8 · Zbl 0455.18010 |
| [3] | Fuchs, L., (Infinite Abelian Groups, Vol. 2 (1973), Academic Press: Academic Press New York) · Zbl 0253.20055 |
| [4] | Lady, E. L., Grothendieck rings for certain categories of quasihomomorphisms of torsion free modules over dedekind domains, J. Algebra, 78, 273-281 (1982) · Zbl 0509.13013 |
| [5] | Lady, E. L., A seminar on splitting rings for torsion free modules over dedekind domains, (Abelian Group Theory. Abelian Group Theory, Lecture Notes in Mathematics, Vol. 1006 (1983), Springer-Verlag: Springer-Verlag New York/Berlin), 1-48 · Zbl 0567.13004 |
| [6] | Lady, E. L., Splitting fields for torsion-free modules over discrete valuation rings, III, J. Algebra, 66, 307-320 (1980) · Zbl 0455.13004 |
| [7] | Warfield, R. B., Homomorphisms and duality for torsion-free groups, Math. Z., 107, 189-200 (1968) · Zbl 0169.03602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
