De Aldama Sanchez R., Chaînes et dépendance, Thèse Lyon 1, septembre 2009.
L. Bélair and F. Point, Quantifier elimination in valued Ore modules, J. Symb. Logic 75 (2010) 1007-1034. MR2723780. Corrigendum to: Quantifier to elimation in valued Ore modules, J.Symb.Logic77 (2012), no. 2, 727-728. MR2963033
L. Bélair and F. Point, Separably closed fields and contractive Ore modules, J. Symb. Log. 80 (2015), no. 4, 1315–1338, DOI 10.1017/jsl.2015.42. MR3436370
L. Bélair, A. Macintyre, and T. Scanlon, Model theory of the Frobenius on the Witt vectors, Amer. J. Math. 129 (2007), no. 3, 665–721, DOI 10.1353/ajm.2007.0018. MR2325101
Z. Chatzidakis, La représentation en termes de faisceaux des modèles de la théorie élémentaire de la multiplication des entiers naturels (French), Model theory and arithmetic (Paris, 1979), Lecture Notes in Math., vol. 890, Springer, Berlin-New York, 1981, pp. 90–110. MR644997
P. M. Cohn, Skew fields: Theory of general division rings, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995. MR1349108
P. F. Conrad, Embedding theorems for abelian groups with valuations, Amer.J.Math.75 (1953), 1–29, DOI 10.2307/2372611. MR0053933
L. van den Dries, Quantifier elimination for linear formulas over ordered and valued fields, Proceedings of the Model Theory Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980), Bull. Soc. Math. Belg. Sér. B 33 (1981), no. 1, 19–31. MR620959
L. van den Dries, Elimination theory for the ring of algebraic integers, J.ReineAngew.Math. 388 (1988), 189–205, DOI 10.1515/crll.1988.388.189. MR944190
L. van den Dries and A. Macintyre, The logic of Rumely’s local-global principle, J.Reine Angew. Math. 407 (1990), 33–56. MR1048527
F. Delon and P. Simonetta, Abelian C-minimal valued groups, Ann. Pure Appl. Logic 168 (2017), no. 9, 1729–1782, DOI 10.1016/j.apal.2017.03.008. MR3659410
Ju. L. Eršov, Decidability of the elementary theory of relatively complemented lattices and of the theory of filters (Russian), Algebra i Logika Sem. 3 (1964), no. 3, 17–38. MR0180490
I. Fleischer, Maximality and ultracompleteness in normed modules, Proc. Amer. Math. Soc. 9 (1958), 151–157, DOI 10.2307/2033414. MR0093693
L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. MR1794715
S. Garavaglia, Direct product decomposition of theories of modules, J. Symbolic Logic 44 (1979), no. 1, 77–88, DOI 10.2307/2273705. MR523490
N. Garcia-Fritz and H. Pasten, Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic, J. Number Theory 156 (2015), 368–393, DOI 10.1016/j.jnt.2015.04.018. MR3360345
L. Gregory, Decidability for theories of modules over valuation domains, J. Symb. Log. 80 (2015), no. 2, 684–711, DOI 10.1017/jsl.2014.1. MR3377362
Guignot F., Théorie des modèles des groupes abéliens valués, Thèse Paris 7, novembre 2016.
M. Henriksen, On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179–184. MR0047928
M. Henriksen, On the prime ideals of the ring of entire functions, PacificJ.Math.3 (1953), 711–720. MR0059479
W. Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR1221741
T. G. Kucera and M. Prest, Imaginary modules, J. Symbolic Logic 57 (1992), no. 2, 698–723, DOI 10.2307/2275302. MR1169204
S. Kochen, The model theory of local fields, ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974), Lecture Notes in Math., vol. 499, Springer, Berlin, 1975, pp. 384–425. MR0568318
M. Lazard, Groupes analytiques p-adiques (French), Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603. MR0209286
L’Innocente S., Point F., Bézout domains and lattice-valued modules, preprint, arXiv:1604.05922.
S. L’Innocente, C. Toffalori, and G. Puninski, On the decidability of the theory of modules over the ring of algebraic integers, Ann. Pure Appl. Logic 168 (2017), no. 8, 1507–1516, DOI 10.1016/j.apal.2017.02.003. MR3650351
S. L’Innocente, F. Point, G. Puninski, and C. Toffalori, The Ziegler spectrum of the ring of entire complex valued functions, J.Symb.Logic84 (2019), no. 1, 160–177.
L. Lipshitz and T. Pheidas, An analogue of Hilbert’s tenth problem for p-adic entire functions, J. Symbolic Logic 60 (1995), no. 4, 1301–1309, DOI 10.2307/2275889. MR1367211
L. Lipshitz and D. Saracino, The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc. 38 (1973), 381–387, DOI 10.2307/2039295. MR0439624
D. Macpherson and C. Steinhorn, On variants of o-minimality, Ann. Pure Appl. Logic 79 (1996), no. 2, 165–209, DOI 10.1016/0168-0072(95)00037-2. MR1396850
N. Mariaule, The field of p-adic numbers with a predicate for the powers of an integer, J. Symb. Log. 82 (2017), no. 1, 166–182, DOI 10.1017/jsl.2016.22. MR3631281
J. Ohm, Semi-valuations and groups of divisibility, Canad. J. Math. 21 (1969), 576–591, DOI 10.4153/CJM-1969-065-9. MR0242819
Onay G., Modules valués, Thèse Paris 7, 2011.
K. Pal, Multiplicative valued difference fields, J. Symbolic Logic 77 (2012), no. 2, 545–579, DOI 10.2178/jsl/1333566637. MR2963021
T. Pheidas and K. Zahidi, Undecidability of existential theories of rings and fields: a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 49–105, DOI 10.1090/conm/270/04369. MR1802009
M. Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR933092
A. Prestel and J. Schmid, Existentially closed domains with radical relations: an axiomatization of the ring of algebraic integers, J.ReineAngew.Math.407 (1990), 178–201, DOI 10.1515/crll.1990.407.178. MR1048534
A. Prestel and J. Schmid, Decidability of the rings of real algebraic and p-adic algebraic integers, J.ReineAngew.Math.414 (1991), 141–148, DOI 10.1515/crll.1991.414.141. MR1092628
G. Puninski, V. Puninskaya, and C. Toffalori, Decidability of the theory of modules over commutative valuation domains, Ann. Pure Appl. Logic 145 (2007), no. 3, 258–275, DOI 10.1016/j.apal.2006.09.002. MR2286415
G. Puninski and C. Toffalori, Some model theory of modules over Bézout domains. The width, J. Pure Appl. Algebra 219 (2015), no. 4, 807–829, DOI 10.1016/j.jpaa.2014.04.031. MR3282111
T. Rohwer, Valued difference fields as modules over twisted polynomial rings, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR2705045
W. Rudin, Analyse réelle et complexe (French), Masson et Cie, Éditeurs, Paris, 1975. Traduit de l’anglais par N. Dhombres et F. Hoffman. MR0389461
W. Rump, Abelian lattice-ordered groups and a characterization of the maximal spectrum of a Prüfer domain, J. Pure Appl. Algebra 218 (2014), no. 12, 2204–2217, DOI 10.1016/j.jpaa.2014.03.011. MR3227300
W. Rump and Y. C. Yang, Jaffard-Ohm correspondence and Hochster duality, Bull. Lond. Math. Soc. 40 (2008), no. 2, 263–273, DOI 10.1112/blms/bdn006. MR2414785
P. H. Schmitt, Undecidable theories of valuated abelian groups, Mém.Soc.Math.France (N.S.) 16 (1984), 67–76. Logic (Paris, 1983). MR792494
P. H. Schmitt, Decidable theories of valuated abelian groups, Logic colloquium ’84 (Manchester, 1984), Stud. Logic Found. Math., vol. 120, North-Holland, Amsterdam, 1986, pp. 245–276, DOI 10.1016/S0049-237X(08)70466-X. MR861428
D. Zelinsky, Topological characterization of fields with valuations, Bull. Amer. Math. Soc. 54 (1948), 1145–1150, DOI 10.1090/S0002-9904-1948-09141-8. MR0028303
M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213, DOI 10.1016/0168-0072(84)90014-9. MR739577
