×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.