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When every ideal is $\phi$-P-flat

Yıl 2023, , 708 - 720, 30.05.2023
https://doi.org/10.15672/hujms.1148258

Öz

Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is called $\phi$-P-flat if $x \in \Ann(s)M$ for every non-nilpotent element $s \in R$ and $x\in M$ such that $sx=0$. In this paper, we introduce and study the class of $\phi$-PF-rings, i.e., rings in which all ideals are $\phi$-P-flat. Among other results, the transfer of the $\phi$-PF-ring to the amalgamation is investigated. Several examples which delineate the concepts and results are provided.

Destekleyen Kurum

National Research Foundation of Korea

Proje Numarası

2021R1I1A3047469

Kaynakça

  • [1] D. F. Anderson and A. Badawi, On $\phi$-Prüfer rings and $\phi$-Bézout rings, Houston J. Math 30 (2), 331–343, 2004.
  • [2] D. D. Anderson and M. Winders, Idealization of a module, J. Comm. Algebra 1 (1), 3–56, 2009.
  • [3] G. Artico and U. Marconi, On the compactness of minimal spectrum, Rend. Sem. Mat. Univ. Padova 56, 79–84, 1976.
  • [4] A. Badawi, On divided commutative rings, Comm. Algebra 27 (3), 1465–1474, 1999.
  • [5] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extensions defined by Prüfer condi- tions, J. Pure Appl. Algebra 214 (1), 53–60, 2010.
  • [6] G. W. Chang and H. Kim, Prüfer rings in a certain pullback, Comm. Algebra, to appear, 2022, doi: 10.1080/00927872.2022.2149766.
  • [7] F. Cheniour and N. Mahdou, When every principal ideal is flat, Port. Math. (N.S.), 70 (1), 51–58, 2011.
  • [8] F. Couchot, Flat modules over valuation rings, J. Pure Appl. Algebra 211 (1), 235– 247, 2007.
  • [9] M. D’Anna, A construction of Gorenstein rings, J. Algebra 306 (2), 507–519, 2006.
  • [10] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in: Commutative Algebra and Applications, Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications, Fez, Morocco, 2008, pp. 155-172, W. de Gruyter Publisher, Berlin, 2009.
  • [11] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in amalgamated algebras along an ideal, J. Pure Appl. Algebra 214 (9), 1633–1641, 2010.
  • [12] M. D’Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra 44, 1836–1851, 2016.
  • [13] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
  • [14] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
  • [15] A. El Khalfi, H. Kim and N. Mahdou, Amalgamation extension in commutative ring theory, a survey, Moroccan Journal of Algebra and Geometry with Applications 1 (1), 139–182, 2022.
  • [16] S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics 1371, Berlin: Spring-Verlag, 1989.
  • [17] J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York Basel, 1988.
  • [18] S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (10), 3937–3953, 2004.
  • [19] H. Kim and F. Wang, On $\phi$-strong Mori rings, Houston J. Math. 38 (2), 359–371, 2012.
  • [20] J.J. Rotman, An Introduction to Homological Algebra, Springer, New York, 2009.
  • [21] F.Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications 22, Springer, Singapore, 2016.
  • [22] W. Zhao, On $\phi$-flat modules and $\phi$-Prüfer rings, J. Korean Math. Soc. 55 (5), 1221– 1233, 2018.
  • [23] W. Zhao, F. Wang and G. Tang, On $\phi$-von Neumann regular rings, J. Korean Math. Soc. 50 (1), 219–229, 2013.
Yıl 2023, , 708 - 720, 30.05.2023
https://doi.org/10.15672/hujms.1148258

Öz

Proje Numarası

2021R1I1A3047469

Kaynakça

  • [1] D. F. Anderson and A. Badawi, On $\phi$-Prüfer rings and $\phi$-Bézout rings, Houston J. Math 30 (2), 331–343, 2004.
  • [2] D. D. Anderson and M. Winders, Idealization of a module, J. Comm. Algebra 1 (1), 3–56, 2009.
  • [3] G. Artico and U. Marconi, On the compactness of minimal spectrum, Rend. Sem. Mat. Univ. Padova 56, 79–84, 1976.
  • [4] A. Badawi, On divided commutative rings, Comm. Algebra 27 (3), 1465–1474, 1999.
  • [5] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extensions defined by Prüfer condi- tions, J. Pure Appl. Algebra 214 (1), 53–60, 2010.
  • [6] G. W. Chang and H. Kim, Prüfer rings in a certain pullback, Comm. Algebra, to appear, 2022, doi: 10.1080/00927872.2022.2149766.
  • [7] F. Cheniour and N. Mahdou, When every principal ideal is flat, Port. Math. (N.S.), 70 (1), 51–58, 2011.
  • [8] F. Couchot, Flat modules over valuation rings, J. Pure Appl. Algebra 211 (1), 235– 247, 2007.
  • [9] M. D’Anna, A construction of Gorenstein rings, J. Algebra 306 (2), 507–519, 2006.
  • [10] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in: Commutative Algebra and Applications, Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications, Fez, Morocco, 2008, pp. 155-172, W. de Gruyter Publisher, Berlin, 2009.
  • [11] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in amalgamated algebras along an ideal, J. Pure Appl. Algebra 214 (9), 1633–1641, 2010.
  • [12] M. D’Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra 44, 1836–1851, 2016.
  • [13] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
  • [14] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
  • [15] A. El Khalfi, H. Kim and N. Mahdou, Amalgamation extension in commutative ring theory, a survey, Moroccan Journal of Algebra and Geometry with Applications 1 (1), 139–182, 2022.
  • [16] S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics 1371, Berlin: Spring-Verlag, 1989.
  • [17] J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York Basel, 1988.
  • [18] S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (10), 3937–3953, 2004.
  • [19] H. Kim and F. Wang, On $\phi$-strong Mori rings, Houston J. Math. 38 (2), 359–371, 2012.
  • [20] J.J. Rotman, An Introduction to Homological Algebra, Springer, New York, 2009.
  • [21] F.Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications 22, Springer, Singapore, 2016.
  • [22] W. Zhao, On $\phi$-flat modules and $\phi$-Prüfer rings, J. Korean Math. Soc. 55 (5), 1221– 1233, 2018.
  • [23] W. Zhao, F. Wang and G. Tang, On $\phi$-von Neumann regular rings, J. Korean Math. Soc. 50 (1), 219–229, 2013.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Hwankoo Kım 0000-0002-3563-077X

Najib Mahdou 0000-0001-6353-1114

El Houssaine Oubouhou 0000-0002-5344-4153

Proje Numarası 2021R1I1A3047469
Yayımlanma Tarihi 30 Mayıs 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Kım, H., Mahdou, N., & Oubouhou, E. H. (2023). When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics, 52(3), 708-720. https://doi.org/10.15672/hujms.1148258
AMA Kım H, Mahdou N, Oubouhou EH. When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics. Mayıs 2023;52(3):708-720. doi:10.15672/hujms.1148258
Chicago Kım, Hwankoo, Najib Mahdou, ve El Houssaine Oubouhou. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics 52, sy. 3 (Mayıs 2023): 708-20. https://doi.org/10.15672/hujms.1148258.
EndNote Kım H, Mahdou N, Oubouhou EH (01 Mayıs 2023) When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics 52 3 708–720.
IEEE H. Kım, N. Mahdou, ve E. H. Oubouhou, “When every ideal is $\phi$-P-flat”, Hacettepe Journal of Mathematics and Statistics, c. 52, sy. 3, ss. 708–720, 2023, doi: 10.15672/hujms.1148258.
ISNAD Kım, Hwankoo vd. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics 52/3 (Mayıs 2023), 708-720. https://doi.org/10.15672/hujms.1148258.
JAMA Kım H, Mahdou N, Oubouhou EH. When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics. 2023;52:708–720.
MLA Kım, Hwankoo vd. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics, c. 52, sy. 3, 2023, ss. 708-20, doi:10.15672/hujms.1148258.
Vancouver Kım H, Mahdou N, Oubouhou EH. When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics. 2023;52(3):708-20.