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Some results on relative dual Baer property

Year 2020, , 259 - 267, 06.09.2020
https://doi.org/10.13069/jacodesmath.790751

Abstract

Let $R$ be a ring.
In this article, we introduce and study relative dual Baer property.
We characterize $R$-modules $M$ which are $R_R$-dual Baer, where $R$ is a commutative principal ideal domain.
It is shown that over a right noetherian right hereditary ring $R$, an $R$-module $M$ is $N$-dual Baer for
all $R$-modules $N$ if and only if $M$ is an injective $R$-module.
It is also shown that for $R$-modules $M_1$, $M_2$, $\ldots$, $M_n$ such that $M_i$ is $M_j$-projective for all
$i > j \in \{1,2,\ldots, n\}$, an $R$-module $N$ is $\bigoplus_{i=1}^nM_i$-dual Baer if and only if $N$ is
$M_i$-dual Baer for all $i\in \{1,2,\ldots,n\}$.
We prove that an $R$-module $M$ is dual Baer if and only if $S=End_R(M)$ is a Baer ring
and $IM=r_M(l_S(IM))$ for every right ideal $I$ of $S$.

References

  • [1] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, vol. 13, Springer–Verlag, New York 1992.
  • [2] E. P. Armendariz, A note on extensions of Baer and P.P.–rings, J. Austral. Math. Soc. 18(4) (1974) 470–473.
  • [3] G. F. Birkenmeier, J. Y. Kim, J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159(1) (2001) 25–42.
  • [4] K. A. Byrd, Rings whose quasi-injective modules are injective, Proc. Amer. Math. Soc. 33(2) (1972) 235–240.
  • [5] S. M. Khuri, Baer endomorphism rings and closure operators, Canad. J. Math. 30(5) (1978) 1070– 1078.
  • [6] I. Kaplansky, Rings of Operators, W. A. Benjamin Inc., New York-Amsterdam 1968.
  • [7] G. Lee, S. T. Rizvi, C. S. Roman, Rickart modules, Comm. Algebra 38(11) (2010) 4005–4027.
  • [8] G. Lee, S. T. Rizvi, C. S. Roman, Dual Rickart modules, Comm. Algebra 39(11) (2011) 4036–4058.
  • [9] S. H. Mohamed, B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge University Press 1990.
  • [10] S. T. Rizvi, C. S. Roman, Baer and quasi-Baer modules, Comm. Algebra 32(1) (2004) 103–123.
  • [11] S. T. Rizvi, C. S. Roman, Baer property of modules and applications, Advances in Ring Theory (2005) 225–241.
  • [12] D. W. Sharpe, P. Vámos, Injective Modules, Cambridge University Press, Cambridge 1972.
  • [13] Y. Talebi, N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra 30(3) (2002) 1449–1460.
  • [14] D. K. Tütüncü and R. Tribak, On dual Baer modules, Glasgow Math. J. 52(2) (2010) 261–269.
  • [15] D. K. Tütüncü, P. F. Smith, S. E. Toksoy, On dual Baer modules, Contemp. Math. 609 (2014) 173–184.
  • [16] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia 1991.
Year 2020, , 259 - 267, 06.09.2020
https://doi.org/10.13069/jacodesmath.790751

Abstract

References

  • [1] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, vol. 13, Springer–Verlag, New York 1992.
  • [2] E. P. Armendariz, A note on extensions of Baer and P.P.–rings, J. Austral. Math. Soc. 18(4) (1974) 470–473.
  • [3] G. F. Birkenmeier, J. Y. Kim, J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159(1) (2001) 25–42.
  • [4] K. A. Byrd, Rings whose quasi-injective modules are injective, Proc. Amer. Math. Soc. 33(2) (1972) 235–240.
  • [5] S. M. Khuri, Baer endomorphism rings and closure operators, Canad. J. Math. 30(5) (1978) 1070– 1078.
  • [6] I. Kaplansky, Rings of Operators, W. A. Benjamin Inc., New York-Amsterdam 1968.
  • [7] G. Lee, S. T. Rizvi, C. S. Roman, Rickart modules, Comm. Algebra 38(11) (2010) 4005–4027.
  • [8] G. Lee, S. T. Rizvi, C. S. Roman, Dual Rickart modules, Comm. Algebra 39(11) (2011) 4036–4058.
  • [9] S. H. Mohamed, B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge University Press 1990.
  • [10] S. T. Rizvi, C. S. Roman, Baer and quasi-Baer modules, Comm. Algebra 32(1) (2004) 103–123.
  • [11] S. T. Rizvi, C. S. Roman, Baer property of modules and applications, Advances in Ring Theory (2005) 225–241.
  • [12] D. W. Sharpe, P. Vámos, Injective Modules, Cambridge University Press, Cambridge 1972.
  • [13] Y. Talebi, N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra 30(3) (2002) 1449–1460.
  • [14] D. K. Tütüncü and R. Tribak, On dual Baer modules, Glasgow Math. J. 52(2) (2010) 261–269.
  • [15] D. K. Tütüncü, P. F. Smith, S. E. Toksoy, On dual Baer modules, Contemp. Math. 609 (2014) 173–184.
  • [16] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia 1991.
There are 16 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Tayyebeh Amouzegar This is me 0000-0002-0600-5326

Rachid Tribak 0000-0001-8400-4321

Publication Date September 6, 2020
Published in Issue Year 2020

Cite

APA Amouzegar, T., & Tribak, R. (2020). Some results on relative dual Baer property. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(3), 259-267. https://doi.org/10.13069/jacodesmath.790751
AMA Amouzegar T, Tribak R. Some results on relative dual Baer property. Journal of Algebra Combinatorics Discrete Structures and Applications. September 2020;7(3):259-267. doi:10.13069/jacodesmath.790751
Chicago Amouzegar, Tayyebeh, and Rachid Tribak. “Some Results on Relative Dual Baer Property”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, no. 3 (September 2020): 259-67. https://doi.org/10.13069/jacodesmath.790751.
EndNote Amouzegar T, Tribak R (September 1, 2020) Some results on relative dual Baer property. Journal of Algebra Combinatorics Discrete Structures and Applications 7 3 259–267.
IEEE T. Amouzegar and R. Tribak, “Some results on relative dual Baer property”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 3, pp. 259–267, 2020, doi: 10.13069/jacodesmath.790751.
ISNAD Amouzegar, Tayyebeh - Tribak, Rachid. “Some Results on Relative Dual Baer Property”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/3 (September 2020), 259-267. https://doi.org/10.13069/jacodesmath.790751.
JAMA Amouzegar T, Tribak R. Some results on relative dual Baer property. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:259–267.
MLA Amouzegar, Tayyebeh and Rachid Tribak. “Some Results on Relative Dual Baer Property”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 3, 2020, pp. 259-67, doi:10.13069/jacodesmath.790751.
Vancouver Amouzegar T, Tribak R. Some results on relative dual Baer property. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(3):259-67.