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.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.