Geodesic
אn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL*
Glasgow mathematical journal, Tome 55 (2013) no. 2, pp. 369-380

Voir la notice de l'article provenant de la source Cambridge University Press

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the אn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large אn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.
DOI : 10.1017/S0017089512000614
Mots-clés : 20A15, 20K10, 20K20, 20K21, 20K30, 13B10, 13L05 
GÖBEL, RÜDIGER; SHELAH, SAHARON; STRÜNGMANN, LUTZ. אn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL*. Glasgow mathematical journal, Tome 55 (2013) no. 2, pp. 369-380. doi: 10.1017/S0017089512000614
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     title = {אn-FREE {MODULES} {OVER} {COMPLETE} {DISCRETE} {VALUATION} {DOMAINS} {WITH} {ALMOST} {TRIVIAL} {DUAL*}},
     journal = {Glasgow mathematical journal},
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[1] 1.Corner, A. L. S., Every countable reduced torsion-free ring is an endomorphism ring, Proc. Lond. Math. Soc. 13 (3) (1963), 687–710. Google Scholar | DOI

[2] 2.Corner, A. L. S. and Göbel, R., Prescribing endomorphism algebras – a unified treatment, Proc. Lond. Math. Soc. 50 (3) (1985), 447–479. Google Scholar | DOI

[3] 3.Dugas, M. and Göbel, R., Almost Σ-cyclic abelian p-groups in L, in Abelian groups and modules, proceedings of an international conference, Udine, CISM Courses and Lectures, vol. 287 (Springer, New York, 1984), 87–105. Google Scholar

[4] 4.Eklof, P. C. and Mekler, A. H., Almost free modules, Revised ed. (North–Holland, New York, 2002). Google Scholar | DOI

[5] 5.Fuchs, L., Infinite abelian groups, vol. 1 & 2 (Academic Press, New York, 1970, 1973). Google Scholar

[6] 6.Göbel, R., Herden, D. and Shelah, S., Prescribing endomorphism algebras of -free modules (to be submitted). Google Scholar

[7] 7.Göbel, R. and May, W., Independence in completions and endomorphism algebras, Forum Math. 1 (1989), 215–226. Google Scholar

[8] 8.Göbel, R. and Paras, A., Splitting off free summands of torsion-free modules over complete DVRs, Glasgow Math. J. 44 (2002), 349–351. Google Scholar | DOI

[9] 9.Göbel, R. and Shelah, S., -free modules with trivial dual, Results Math. 54 (2009), 53–64. Google Scholar

[10] 10.Göbel, R. and Trlifaj, J., Endomorphism algebras and approximations of modules, Expositions in Mathematics, vol. 41 (Walter de Gruyter Verlag, Berlin, Germany, 2006). Google Scholar | DOI

[11] 11.Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, MI, 1971). Google Scholar

[12] 12.Krylov, P. A. and Tuganbaev, A. A., Modules over discrete valuation domains, Expositions in Mathematics, vol 43 (Walter de Gruyter Verlag, Berlin, Germany, 2008). Google Scholar | DOI

[13] 13.Shelah, S., -free abelian groups with no non-zero homomorphisms to ℤ, CUBO Math. J. 9 (2007), 59–79. Google Scholar

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