On Bhargava rings

Mohamed Mahmoud Chems-Eddin; Omar Ouzzaouit; Ali Tamoussit

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 2, page 181-195
  • ISSN: 0862-7959

Abstract

top
Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be 𝔹 x ( D ) : = { f ∈ K [ X ] : for all a ∈ D , f ( x X + a ) ∈ D [ X ] } . In fact, 𝔹 x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of 𝔹 x ( D ) under localization. In particular, we prove that 𝔹 x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D such that 𝔹 x ( D ) is locally free, or at least faithfully flat (or flat) as a D -module (or D [ X ] -module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which 𝔹 x ( D ) is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.

How to cite

top

Chems-Eddin, Mohamed Mahmoud, Ouzzaouit, Omar, and Tamoussit, Ali. "On Bhargava rings." Mathematica Bohemica 148.2 (2023): 181-195. <http://eudml.org/doc/299529>.

@article{Chems2023,
abstract = {Let $D$ be an integral domain with the quotient field $K$, $X$ an indeterminate over $K$ and $x$ an element of $D$. The Bhargava ring over $D$ at $x$ is defined to be $\mathbb \{B\}_x(D):=\lbrace f\in K[X]\colon \text\{for all\}\ a\in D,\ f(xX+a)\in D[X]\rbrace $. In fact, $\mathbb \{B\}_x(D)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of $\mathbb \{B\}_x(D)$ under localization. In particular, we prove that $\mathbb \{B\}_x(D)$ behaves well under localization at prime ideals of $D$, when $D$ is a locally finite intersection of localizations. We also attempt a classification of integral domains $D$ such that $\mathbb \{B\}_x(D)$ is locally free, or at least faithfully flat (or flat) as a $D$-module (or $D[X]$-module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which $\mathbb \{B\}_x(D)$ is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.},
author = {Chems-Eddin, Mohamed Mahmoud, Ouzzaouit, Omar, Tamoussit, Ali},
journal = {Mathematica Bohemica},
keywords = {Bhargava ring; localization; (locally) essential domain; locally free module; (faithfully) flat module; Krull dimension},
language = {eng},
number = {2},
pages = {181-195},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Bhargava rings},
url = {http://eudml.org/doc/299529},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Chems-Eddin, Mohamed Mahmoud
AU - Ouzzaouit, Omar
AU - Tamoussit, Ali
TI - On Bhargava rings
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 2
SP - 181
EP - 195
AB - Let $D$ be an integral domain with the quotient field $K$, $X$ an indeterminate over $K$ and $x$ an element of $D$. The Bhargava ring over $D$ at $x$ is defined to be $\mathbb {B}_x(D):=\lbrace f\in K[X]\colon \text{for all}\ a\in D,\ f(xX+a)\in D[X]\rbrace $. In fact, $\mathbb {B}_x(D)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of $\mathbb {B}_x(D)$ under localization. In particular, we prove that $\mathbb {B}_x(D)$ behaves well under localization at prime ideals of $D$, when $D$ is a locally finite intersection of localizations. We also attempt a classification of integral domains $D$ such that $\mathbb {B}_x(D)$ is locally free, or at least faithfully flat (or flat) as a $D$-module (or $D[X]$-module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which $\mathbb {B}_x(D)$ is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.
LA - eng
KW - Bhargava ring; localization; (locally) essential domain; locally free module; (faithfully) flat module; Krull dimension
UR - http://eudml.org/doc/299529
ER -

References

top
  1. Alrasasi, I., Izelgue, L., 10.1080/00927870902922968, Commun. Algebra 38 (2010), 1385-1400. (2010) Zbl1198.13019MR2656583DOI10.1080/00927870902922968
  2. Al-Rasasi, I., Izelgue, L., 10.1007/978-3-319-74195-6_2, Homological and Combinatorial Methods in Algebra Springer Proceedings in Mathematics & Statistics 228. Springer, Cham (2018), 9-26. (2018) Zbl1402.13019MR3778007DOI10.1007/978-3-319-74195-6_2
  3. Anderson, D. D., Anderson, D. F., Generalized GCD domains, Comment. Math. Univ. St. Pauli 28 (1980), 215-221. (1980) Zbl0434.13001MR0578675
  4. Anderson, D. D., Anderson, D. F., Zafrullah, M., Rings between D [ X ] and K [ X ] , Houston J. Math. 17 (1991), 109-129. (1991) Zbl0736.13015MR1107192
  5. Bhargava, M., Cahen, P.-J., Yeramian, J., 10.1016/j.jalgebra.2009.04.017, J. Algebra 322 (2009), 1129-1150. (2009) Zbl1177.13051MR2537676DOI10.1016/j.jalgebra.2009.04.017
  6. Bourbaki, N., 10.1007/978-3-540-33976-2, Masson, Paris (1985), French. (1985) Zbl1103.13001MR782296DOI10.1007/978-3-540-33976-2
  7. Brewer, J. W., Heinzer, W. J., 10.1215/S0012-7094-74-04101-5, Duke Math. J. 41 (1974), 1-7. (1974) Zbl0284.13001MR0335486DOI10.1215/S0012-7094-74-04101-5
  8. Cahen, P.-J., Chabert, J.-L., 10.1090/surv/048, Mathematical Surveys Monographs 48. American Mathematical Society, Providence (1997). (1997) Zbl0884.13010MR1421321DOI10.1090/surv/048
  9. El-Baghdadi, S., 10.1080/00927870903114961, Commun. Algebra 38 (2010), 3029-3044. (2010) Zbl1203.13002MR2730293DOI10.1080/00927870903114961
  10. Elliott, J., 10.1515/9783110213188.223, Commutative Algebra and Its Applications Walter de Gruyter, Berlin (2009), 223-237. (2009) Zbl1177.13053MR2606288DOI10.1515/9783110213188.223
  11. Elliott, J., 10.1080/00927872.2010.519366, Commun. Algebra 39 (2011), 4128-4147. (2011) Zbl1247.13022MR2855117DOI10.1080/00927872.2010.519366
  12. Fontana, M., Kabbaj, S., 10.1090/S0002-9939-04-07502-1, Proc. Am. Math. Soc. 132 (2004), 2529-2535. (2004) Zbl1059.13008MR2054776DOI10.1090/S0002-9939-04-07502-1
  13. Gilmer, R., Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics 90. Queen's University, Kingston (1992). (1992) Zbl0804.13001MR1204267
  14. R. W. Gilmer, Jr., 10.1016/0021-8693(66)90025-1, J. Algebra 4 (1966), 331-340. (1966) Zbl0146.26205MR0202749DOI10.1016/0021-8693(66)90025-1
  15. Heinzer, W., 10.4153/CJM-1981-034-8, Can. J. Math. 33 (1981), 400-403. (1981) Zbl0411.13013MR0617630DOI10.4153/CJM-1981-034-8
  16. Heinzer, W., Roitman, M., 10.1016/S0021-8693(03)00462-9, J. Algebra 272 (2004), 435-455. (2004) Zbl1040.13002MR2028066DOI10.1016/S0021-8693(03)00462-9
  17. Heubo-Kwegna, O. A., Olberding, B., Reinhart, A., 10.1016/j.jpaa.2016.05.021, J. Pure Appl. Algebra 220 (2016), 3927-3947. (2016) Zbl1353.13020MR3517563DOI10.1016/j.jpaa.2016.05.021
  18. Hutchins, H. C., Examples of Commutative Rings, Polygonal Publishing House, Washington (1981). (1981) Zbl0492.13001MR0638720
  19. Izelgue, L., Mimouni, A. A., Tamoussit, A., 10.1007/s40840-019-00826-5, Bull. Malays. Math. Sci. Soc. (2) 43 (2020), 2687-2699. (2020) Zbl1437.13030MR4089663DOI10.1007/s40840-019-00826-5
  20. Kim, H., Tamoussit, A., 10.1080/00927872.2021.1960991, Commun. Algebra 50 (2022), 538-555. (2022) MR4375523DOI10.1080/00927872.2021.1960991
  21. Mott, J. L., Zafrullah, M., 10.1007/BF01168446, Manuscr. Math. 35 (1981), 1-26. (1981) Zbl0477.13007MR0627923DOI10.1007/BF01168446
  22. Park, M. H., Tartarone, F., 10.1142/S021949882050098X, J. Algebra Appl. 19 (2020), Article ID 2050098, 14 pages. (2020) Zbl1445.13020MR4114450DOI10.1142/S021949882050098X
  23. E. M. Pirtle, Jr., 10.32917/hmj/1206138662, J. Sci. Hiroshima Univ., Ser. A-I 32 (1968), 441-447. (1968) Zbl0181.04903MR0244221DOI10.32917/hmj/1206138662
  24. E. M. Pirtle, Jr., 10.1090/S0002-9947-1969-0249416-4, Trans. Am. Math. Soc. 144 (1969), 427-439. (1969) Zbl0197.03203MR0249416DOI10.1090/S0002-9947-1969-0249416-4
  25. Tamoussit, A., 10.1007/s40574-021-00281-w, Boll. Unione Mat. Ital. 14 (2021), 513-519. (2021) Zbl1469.13025MR4290350DOI10.1007/s40574-021-00281-w
  26. Tartarone, F., On the Krull dimension of I n t ( D ) when D is a pullback, Commutative Ring Theory Lecture Notes in Pure Applied Mathematics 185. Marcel Dekker, New York (1997), 457-470. (1997) Zbl0899.13024MR1422501
  27. Yeramian, J., Anneaux de Bhargava: Thèse de Doctorat, Université Paul Cézanne, Marseille (2004), French. (2004) MR2102166
  28. Yeramian, J., 10.1081/AGB-120039278, Commun. Algebra 32 (2004), 3043-3069 French. (2004) Zbl1061.13011MR2102166DOI10.1081/AGB-120039278
  29. Yeramian, J., 10.1016/j.jpaa.2008.11.008, J. Pure Appl. Algebra 213 (2009), 1013-1025. (2009) Zbl1162.13007MR2498793DOI10.1016/j.jpaa.2008.11.008
  30. Zafrullah, M., 10.1016/0022-4049(88)90006-0, J. Pure Appl. Algebra 50 (1988), 93-107. (1988) Zbl0656.13020MR0931909DOI10.1016/0022-4049(88)90006-0

NotesEmbed ?

top

You must be logged in to post comments.