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On the decidability of the theory of modules over the ring of algebraic integers. (English) Zbl 1422.03017
Summary: We will prove that the theory of all modules over the ring of algebraic integers is decidable.

03B25 Decidability of theories and sets of sentences
03C60 Model-theoretic algebra
03C98 Applications of model theory
13C11 Injective and flat modules and ideals in commutative rings
Full Text: DOI arXiv
[1] Atiyah, M.; McDonald, I. G., Introduction to commutative algebra, Addison-Wesley Series in Mathematics, (1994), Westview Press
[2] Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, (1996), Springer
[3] van den Dries, L., Elimination theory for the ring of algebraic integers, J. Reine Angew. Math., 388, 189-205, (1988) · Zbl 0659.12021
[4] van den Dries, L.; Macintyre, A., The logic of Rumely’s local-global principle, J. Reine Angew. Math., 407, 33-56, (1990) · Zbl 0703.13021
[5] Ershov, Yu. L., Decidability problems and constructive models, (1980), Moscow · Zbl 0495.03009
[6] Frölich, A.; Sheperdson, J. C., On the factorization of polynomials in a finite number of steps, Math. Z., 62, 331-334, (1955) · Zbl 0064.24902
[7] Fuchs, L.; Salce, L., Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, (2001), American Mathematical Society · Zbl 0973.13001
[8] Greenberg, N.; Hamkins, J. D.; Hirschfeldt, D.; Miller, R., Effective mathematics in the uncountable, Lecture Notes in Logic, vol. 41, (2013), ASL and Cambridge University Press
[9] Gregory, L., Decidability for the theory of modules over valuation domains, J. Symbolic Logic, 80, 684-711, (2015) · Zbl 1408.03005
[10] Janusz, G. J., Algebraic number fields, Graduate Studies in Mathematics, vol. 7, (1996), American Mathematical Society · Zbl 0854.11001
[11] Jensen, C. U.; Lenzing, H., Model theoretic algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, vol. 10, (1989), CRC Press New York · Zbl 0728.03026
[12] Kedlaya, K. S., The algebraic closure of the power series field in positive characteristic, Trans. Amer. Math. Soc., 29, 12, 3461-3470, (2001) · Zbl 1012.12007
[13] Lenstra, A. K., Factoring multivariate polynomials over finite fields, J. Comput. System Sci., 30, 235-248, (1985) · Zbl 0577.12013
[14] Petrović, M., Iterative methods for simultaneous inclusions of polynomial zeros, Lecture Notes in Mathematics, vol. 1387, (1989), Springer
[15] Prest, M., Model theory and modules, London Mathematical Society Lecture Notes Series, vol. 130, (1988), Cambridge University Press · Zbl 0634.03025
[16] Prest, M., Purity, spectra and localization, Encyclopedia of Mathematics and Its Applications, vol. 121, (2009), Cambridge University Press · Zbl 1205.16002
[17] Prestel, A.; Schmid, K., Existentially closed domains with radical relations, J. Reine Angew. Math., 407, 178-201, (1990) · Zbl 0691.12013
[18] Puninski, G., Serial rings, (2001), Kluwer · Zbl 1032.16001
[19] Puninski, G.; Puninskaya, V.; Toffalori, C., Decidability of the theory of modules over commutative valuation domains, Ann. Pure Appl. Logic, 145, 258-275, (2007) · Zbl 1111.03011
[20] Puninski, G.; Toffalori, C., Some model theory of modules over Bézout domains. the width, J. Pure Appl. Algebra, 219, 807-829, (2015) · Zbl 1393.03021
[21] Rabin, M. O., Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc., 93, 341-360, (1960) · Zbl 0156.01201
[22] Rumely, R., Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math., 368, 127-133, (1986) · Zbl 0581.14014
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