Pure-injective modules

Fakhruddin, Syed M.

Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 120-122

Résumé

It is proved that a pure-injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group T, with induced module structures. Using similar techniques, it is also shown that an R-module has its underlying group pure-injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product.All rings considered are commutative with unity and all modules are unitary. Let Mod-R be the category of modules over a ring R. An exact sequence 0 → A → B → C → 0 in Mod-R is pure-exact if, for any N in Mod-R, 0 → A⊗N → B⊗N → C⊗N → 0 is exact. A module M is pure-injective if it has the injective property relative to the class of pure-exact sequences in Mod-R. A module P is FP (finitely presented) if it is the image of a finitely generated free module with a finitely generated kernel. A module M is compact if it carries a Hausdorff compact topology so that M is a topological R-module. Let T denote the circle group—the group of real numbers modulo the integers—and let X* denote the dual module HomZ(X, T) of the module X.

Détail

DOI : 10.1017/S0017089500001853

Bibliographie

[1] 1.Fakhruddin, S. M., Linearly compact modules over Noetherian rings; submitted. Google Scholar

[2] 2.Fuchs, L., Infinite abelian groups (Academic Press, 1970). Google Scholar

[3] 3.Stöhr, K. O., Dualitäten in der Kategorie der lokalkompakten Moduln über einem Dedekindschen Ring, J. Reíne Angew. Math. 239 (1969), 239–255. Google Scholar

[4] 4.Warfield, R. B. Jr, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719. Google Scholar | DOI

[5] 5.Warfield, R. B. Jr, Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167–172. Google Scholar | DOI