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http://dx.doi.org/10.4134/BKMS.b190846

SOME RESULTS ON INTEGER-VALUED POLYNOMIALS OVER MODULES  

Naghipour, Ali Reza (Department of Mathematics Shahrekord University)
Hafshejani, Javad Sedighi (Department of Mathematics Shahrekord University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1165-1176 More about this Journal
Abstract
Let M be a module over a commutative ring R. In this paper, we study Int(R, M), the module of integer-valued polynomials on M over R, and IntM(R), the ring of integer-valued polynomials on R over M. We establish some properties of Krull dimensions of Int(R, M) and IntM(R). We also determine when Int(R, M) and IntM(R) are nontrivial. Among the other results, it is shown that Int(ℤ, M) is not Noetherian module over IntM(ℤ) ∩ Int(ℤ), where M is a finitely generated ℤ-module.
Keywords
Integer-valued polynomial; Noetherian ring; Krull dimension; polynomially torsion-free;
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1 S. H. Man and P. F. Smith, On chains of prime submodules, Israel J. Math. 127 (2002), 131-155. https://doi.org/10.1007/BF02784529   DOI
2 H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986.
3 R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math. 23 (1993), no. 3, 1041-1062. https://doi.org/10.1216/rmjm/1181072540   DOI
4 A. R. Naghipour, Strongly prime submodules, Comm. Algebra 37 (2009), no. 7, 2193-2199. https://doi.org/10.1080/00927870802467239   DOI
5 A. R. Naghipour, Some results on strongly prime submodules, J. Alg. Sys. 1 (2013), no. 2, 79-89.
6 A. R. Naghipour, M. R. Rismanchian, and J. Sedighi Hafshejani, Some results on the integer-valued polynomials over matrix rings, Comm. Algebra 45 (2017), no. 4, 1675-1686. https://doi.org/10.1080/00927872.2016.1222407   DOI
7 W. Narkiewicz, Polynomial Mappings, Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, 1995. https://doi.org/10.1007/BFb0076894
8 D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, London, 1968.
9 A. Ostrowski, Uber ganzwertige Polynome in algebraischen Zahlkorpern, J. Reine Angew. Math. 149 (1919), 117-124.
10 G. Peruginelli and N. J. Werner, Integral closure of rings of integer-valued polynomials on algebras, in Commutative algebra, 293-305, Springer, New York, 2014.
11 G. Peruginelli and N. J. Werner, Non-triviality conditions for integer-valued polynomial rings on algebras, Monatsh. Math. 183 (2017), no. 1, 177-189. https://doi.org/10.1007/s00605-016-0951-8   DOI
12 G. Polya, Uber ganzwertige Polynome in algebraischen Zahlkorpern, J. Reine Angew. Math. 149 (1919), 97-116.
13 J. Sedighi Hafshejani, A. R. Naghipour, and M. R. Rismanchian, Integer-valued polynomials over block matrix algebras, J. Algebra Appl. 19 (2020), no. 3, 2050053, 17 pp. https://doi.org/10.1142/S021949882050053X   DOI
14 D. E. Rush, The conditions $Int(R){\subseteq}R_S[X]$ and $Int(R_S)=Int(R)_S$ for integer-valued polynomials, J. Pure Appl. Algebra 125 (1998), no. 1-3, 287-303. https://doi.org/10.1016/S0022-4049(96)00107-7   DOI
15 D. E. Rush, Strongly prime submodules, G-submodules and Jacobson modules, Comm. Algebra 40 (2012), no. 4, 1363-1368. https://doi.org/10.1080/00927872.2010.551530   DOI
16 K. Samei, Reduced multiplication modules, Proc. Indian Acad. Sci. Math. Sci. 121 (2011), no. 2, 121-132. https://doi.org/10.1007/s12044-011-0014-y   DOI
17 R. Y. Sharp, Steps in Commutative Algebra, second edition, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
18 N. J. Werner, Integer-valued polynomials on algebras: a survey of recent results and open questions, in Rings, polynomials, and modules, 353-375, Springer, Cham, 2017.
19 S. Yassemi, Weakly associated primes under change of rings, Comm. Algebra 26 (1998), no. 6, 2007-2018. https://doi.org/10.1080/00927879808826256   DOI
20 P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, 48, American Mathematical Society, Providence, RI, 1997.
21 J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156-181. https://doi.org/10.1515/crll.1978.298.156
22 J. Elliott, Integer-valued polynomial rings, t-closure, and associated primes, Comm. Algebra 39 (2011), no. 11, 4128-4147. https://doi.org/10.1080/00927872.2010.519366   DOI
23 J. Elliott, Presentations and module bases of integer-valued polynomial rings, J. Algebra Appl. 12 (2013), no. 1, 1250137, 25 pp. https://doi.org/10.1142/S021949881250137X   DOI
24 J. Elliott, Integer-valued polynomials on commutative rings and modules, Comm. Algebra 46 (2018), no. 3, 1121-1137. https://doi.org/10.1080/00927872.2017.1388811   DOI
25 S. Frisch, Integer-valued polynomials on algebras: a survey, Actes du CIRM 2 (2010), no. 2, 27-32.   DOI
26 T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Berlin-Heidelberg-New York, 1998.
27 S. Frisch, Integer-valued polynomials on algebras, J. Algebra 373 (2013), 414-425. https://doi.org/10.1016/j.jalgebra.2012.10.003   DOI
28 S. Frisch, Polynomial functions on upper triangular matrix algebras, Monatsh. Math. 184 (2017), no. 2, 201-215. https://doi.org/10.1007/s00605-016-1013-y   DOI