Research Article
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Year 2022, , 725 - 736, 01.06.2022
https://doi.org/10.15672/hujms.910906

Abstract

References

  • [1] A. Ait Ouahi, S. Bouchiba and M. El-Arabi, On proper strong Property ($\mathcal A$) for rings and modules, J. Algebra Appl. 19 (12), 2050239, 2020.
  • [2] D.D. Anderson and S. Chun, The set of torsion elements of a module, Commun. Algebra, 42, 1835-1843, 2014.
  • [3] D.D. Anderson and S. Chun, Zero-divisors, torsion elements, and unions of annihilators, Commun. Algebra, 43, 76-83, 2015.
  • [4] D.D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16 (7), 1750143, 2017.
  • [5] D.D. Anderson and S. Chun, McCoy modules and related modules over commutative rings, Commun. Algebra, 45 (6), 2593-2601. 2017.
  • [6] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1, 3-56, 2009.
  • [7] S. Bouchiba, On the vanishing of annihilators of modules, Commun. Algebra, 48 (2), 879-890, 2020.
  • [8] S. Bouchiba and M. El-Arabi, On Property ($\mathcal A$) for modules over direct products of rings, 44 (2), 147-161, 2021.
  • [9] S. Bouchiba, M. El-Arabi and M. Khaloui, When is the idealization$R\ltimes M$ an $\mathcal A$-ring?, J. Algebra Appl. 19 (12), 2050227, 2020.
  • [10] A.Y. Darani, Notes on annihilator conditions in modules over commutative rings, An. Stinnt. Univ. Ovidius Constanta 18, 59-72, 2010.
  • [11] D.E. Dobbs and J. Shapiro, On the strong (A)-ring of Mahdou and Hassani, Mediterr. J. Math. 10, 1995-1997, 2013.
  • [12] C. Faith, Annihilator ideals, associated primes, and Kasch? McCoy commutative rings, Commun. Algebra 19, 1867-1892, 1991.
  • [13] D.F. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Am. Math. Soc. 27 (3), 427-433 1971.
  • [14] R. Gilmer, A. Grams and T. Parker, zero divisors in power series rings, J. für die Reine und Angew. Math. 0278_0279, 145-164, 1975.
  • [15] E. Hashemi, A. Estaji and M. Ziembowski, Answers to some questions concerning rings with Property (A), Proc. Edinburgh Math. Soc. 60, 651-664, 2017.
  • [16] C.Y. Hong, N.K. Kim, Y. Lee and S.T. Ryu, Rings with Property (A) and their extensions, J. Algebra, 315, 612-628, 2007.
  • [17] J.A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, Inc., New York and Basel, 1988.
  • [18] J.A. Huckaba and J.M. Keller, Annihilation of ideals in commutative rings, Pacific J. Math. 83, 375-379, 1979.
  • [19] I. Kaplansky, Commutative Rings, Polygonal Publishing House, Washington, New Jersey, 1994.
  • [20] T.G. Lucas, The diameter of a zero divisor graph, J. Algebra 301, 174-193, 2006.
  • [21] N. Mahdou and A.R. Hassani, On strong (A)-rings, Mediterr. J. Math. 9, 393-402, 2012.
  • [22] N. Mahdou and M. Moutui, On (A)-rings and (SA)-rings issued from amalgamations, Stud. Sci. Math. Hung. 55 (2), 270-279, 2018.
  • [23] R. Mohammadi, A. Moussavi and M. Zahiri, On rings with annihilator condition, Stud. Sci. Math. Hung. 54, 82-96, 2017.
  • [24] Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99, 265-272, 1971.

Countably McCoy rings

Year 2022, , 725 - 736, 01.06.2022
https://doi.org/10.15672/hujms.910906

Abstract

The main goal of this paper is to study the class of countably $\mathcal {A}$-rings (or the countably McCoy rings) introduced by T. Lucas in [The diameter of a zero divisor graph, J. Algebra 301, 174-193, 2006] which turns out to lie properly between the class of $ \mathcal{A}$-rings (or McCoy rings) and the class of total-$\mathcal{A}$-rings. Also, we introduce and investigate the module theoretic version of the countably $\mathcal {A}$-ring notion, namely the countably $\mathcal {A}$-modules. Our focus is shed on the behavior of the countably $\mathcal {A}$-property vis-à-vis the polynomial ring, the power series ring, the idealization and the direct products. Numerous examples are provided to show the limits of the results.

References

  • [1] A. Ait Ouahi, S. Bouchiba and M. El-Arabi, On proper strong Property ($\mathcal A$) for rings and modules, J. Algebra Appl. 19 (12), 2050239, 2020.
  • [2] D.D. Anderson and S. Chun, The set of torsion elements of a module, Commun. Algebra, 42, 1835-1843, 2014.
  • [3] D.D. Anderson and S. Chun, Zero-divisors, torsion elements, and unions of annihilators, Commun. Algebra, 43, 76-83, 2015.
  • [4] D.D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16 (7), 1750143, 2017.
  • [5] D.D. Anderson and S. Chun, McCoy modules and related modules over commutative rings, Commun. Algebra, 45 (6), 2593-2601. 2017.
  • [6] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, 1, 3-56, 2009.
  • [7] S. Bouchiba, On the vanishing of annihilators of modules, Commun. Algebra, 48 (2), 879-890, 2020.
  • [8] S. Bouchiba and M. El-Arabi, On Property ($\mathcal A$) for modules over direct products of rings, 44 (2), 147-161, 2021.
  • [9] S. Bouchiba, M. El-Arabi and M. Khaloui, When is the idealization$R\ltimes M$ an $\mathcal A$-ring?, J. Algebra Appl. 19 (12), 2050227, 2020.
  • [10] A.Y. Darani, Notes on annihilator conditions in modules over commutative rings, An. Stinnt. Univ. Ovidius Constanta 18, 59-72, 2010.
  • [11] D.E. Dobbs and J. Shapiro, On the strong (A)-ring of Mahdou and Hassani, Mediterr. J. Math. 10, 1995-1997, 2013.
  • [12] C. Faith, Annihilator ideals, associated primes, and Kasch? McCoy commutative rings, Commun. Algebra 19, 1867-1892, 1991.
  • [13] D.F. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Am. Math. Soc. 27 (3), 427-433 1971.
  • [14] R. Gilmer, A. Grams and T. Parker, zero divisors in power series rings, J. für die Reine und Angew. Math. 0278_0279, 145-164, 1975.
  • [15] E. Hashemi, A. Estaji and M. Ziembowski, Answers to some questions concerning rings with Property (A), Proc. Edinburgh Math. Soc. 60, 651-664, 2017.
  • [16] C.Y. Hong, N.K. Kim, Y. Lee and S.T. Ryu, Rings with Property (A) and their extensions, J. Algebra, 315, 612-628, 2007.
  • [17] J.A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, Inc., New York and Basel, 1988.
  • [18] J.A. Huckaba and J.M. Keller, Annihilation of ideals in commutative rings, Pacific J. Math. 83, 375-379, 1979.
  • [19] I. Kaplansky, Commutative Rings, Polygonal Publishing House, Washington, New Jersey, 1994.
  • [20] T.G. Lucas, The diameter of a zero divisor graph, J. Algebra 301, 174-193, 2006.
  • [21] N. Mahdou and A.R. Hassani, On strong (A)-rings, Mediterr. J. Math. 9, 393-402, 2012.
  • [22] N. Mahdou and M. Moutui, On (A)-rings and (SA)-rings issued from amalgamations, Stud. Sci. Math. Hung. 55 (2), 270-279, 2018.
  • [23] R. Mohammadi, A. Moussavi and M. Zahiri, On rings with annihilator condition, Stud. Sci. Math. Hung. 54, 82-96, 2017.
  • [24] Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99, 265-272, 1971.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Samir Bouchiba 0000-0001-8051-7074

Abderrazzak Ait Ouahi This is me 0000-0002-7625-1450

Youssef Najem This is me 0000-0001-9868-2481

Publication Date June 1, 2022
Published in Issue Year 2022

Cite

APA Bouchiba, S., Ait Ouahi, A., & Najem, Y. (2022). Countably McCoy rings. Hacettepe Journal of Mathematics and Statistics, 51(3), 725-736. https://doi.org/10.15672/hujms.910906
AMA Bouchiba S, Ait Ouahi A, Najem Y. Countably McCoy rings. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):725-736. doi:10.15672/hujms.910906
Chicago Bouchiba, Samir, Abderrazzak Ait Ouahi, and Youssef Najem. “Countably McCoy Rings”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 725-36. https://doi.org/10.15672/hujms.910906.
EndNote Bouchiba S, Ait Ouahi A, Najem Y (June 1, 2022) Countably McCoy rings. Hacettepe Journal of Mathematics and Statistics 51 3 725–736.
IEEE S. Bouchiba, A. Ait Ouahi, and Y. Najem, “Countably McCoy rings”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 725–736, 2022, doi: 10.15672/hujms.910906.
ISNAD Bouchiba, Samir et al. “Countably McCoy Rings”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 725-736. https://doi.org/10.15672/hujms.910906.
JAMA Bouchiba S, Ait Ouahi A, Najem Y. Countably McCoy rings. Hacettepe Journal of Mathematics and Statistics. 2022;51:725–736.
MLA Bouchiba, Samir et al. “Countably McCoy Rings”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 725-36, doi:10.15672/hujms.910906.
Vancouver Bouchiba S, Ait Ouahi A, Najem Y. Countably McCoy rings. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):725-36.