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

Year 2023, Volume: 52 Issue: 3, 708 - 720, 30.05.2023
https://doi.org/10.15672/hujms.1148258

Abstract

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.

Supporting Institution

National Research Foundation of Korea

Project Number

2021R1I1A3047469

References

  • [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.
Year 2023, Volume: 52 Issue: 3, 708 - 720, 30.05.2023
https://doi.org/10.15672/hujms.1148258

Abstract

Project Number

2021R1I1A3047469

References

  • [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.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Hwankoo Kım 0000-0002-3563-077X

Najib Mahdou 0000-0001-6353-1114

El Houssaine Oubouhou 0000-0002-5344-4153

Project Number 2021R1I1A3047469
Publication Date May 30, 2023
Published in Issue Year 2023 Volume: 52 Issue: 3

Cite

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. May 2023;52(3):708-720. doi:10.15672/hujms.1148258
Chicago Kım, Hwankoo, Najib Mahdou, and El Houssaine Oubouhou. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics 52, no. 3 (May 2023): 708-20. https://doi.org/10.15672/hujms.1148258.
EndNote Kım H, Mahdou N, Oubouhou EH (May 1, 2023) When every ideal is $\phi$-P-flat. Hacettepe Journal of Mathematics and Statistics 52 3 708–720.
IEEE H. Kım, N. Mahdou, and E. H. Oubouhou, “When every ideal is $\phi$-P-flat”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, pp. 708–720, 2023, doi: 10.15672/hujms.1148258.
ISNAD Kım, Hwankoo et al. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics 52/3 (May 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 et al. “When Every Ideal Is $\phi$-P-Flat”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 3, 2023, pp. 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.