Research Article
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Year 2024, Volume: 36 Issue: 36, 73 - 88, 12.07.2024
https://doi.org/10.24330/ieja.1411145

Abstract

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1992.
  • N. V. Dung, D.V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman Scientific & Technical, 1994.
  • Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755-779.
  • P. A. Guil Asensio and A. K. Srivastava, Automorphism-invariant modules satisfy the exchange property, J. Algebra, 388 (2013), 101-106.
  • P. A. Guil Asensio and A. K. Srivastava, Automorphism-invariant modules, in: Noncommutative Rings and Their Applications, Contemp. Math., vol. 634 (2015), 19-30.
  • P. A. Guil Asensio, D. K. Tutuncu and A. K. Srivastava, Modules invariant under automorphisms of their covers and envelopes, Israel J. Math., 206 (2015), 457-482.
  • R. E. Johnson and E. T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc., 36 (1961), 260-268.
  • S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, Cambridge University Press, Cambridge, 1990.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  • S. Singh and Y. Al-Shania , Quasi-injective multiplication modules, Comm. Algebra, 28(7) (2000), 3329-3334.
  • S. Singh and A. K. Srivastava, Rings of invariant module type and automorphism-invariant modules, in: Ring Theory and Its Applications , Contemp. Math., vol. 609 (2014), 299-311.
  • P. F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel), 50(3) (1988), 223-235.
  • P. F. Smith, Multiplication modules and projective modules, Period. Math. Hungar., 29(2) (1994), 163-168.
  • A. K. Srivastava, A. A. Tuganbaev and P. A. Guil Asensio, Invariance of Modules Under Automorphisms of Their Envelopes and Covers, Cambridge University Press, Cambridge, 2021.
  • B. Stenstrom, Rings of Quotients, Springer-Verlag, 1975.
  • H. Tachikawa, On left QF-3 rings, Paci c J. Math., 32 (1970), 255-268.
  • L. V. Thuyet, P. Dan and and T. C. Quynh, Modules which are invariant under idempotents of their envelopes, Colloq. Math., 143 (2016), 237-250.
  • A. A. Tuganbaev, Multiplication modules over noncommutative rings, Sb. Math., 194(11-12) (2003), 1837-1864.
  • A. A. Tuganbaev, Multiplication modules, J. Math. Sci. (N.Y.), 123(2) (2004), 3839-3905.
  • A. A. Tuganbaev, Multiplication modules and ideals, J. Math. Sci. (N.Y.), 136(4) (2006), 4116-4130.
  • A. A. Tuganbaev, Automorphism-invariant modules, J. Math. Sci. (N.Y.), 206 (2015), 694-698.
  • A. A. Tuganbaev, Automorphism-invariant non-singular rings and modules, J. Algebra, 485 (2017), 247-253.
  • R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • H. P. Yu, On quasi-duo rings, Glasgow Math. J., 37 (1995), 21-31.

On Automorphism-invariant multiplication modules over a noncommutative ring

Year 2024, Volume: 36 Issue: 36, 73 - 88, 12.07.2024
https://doi.org/10.24330/ieja.1411145

Abstract

One of the important classes of modules is the class of multiplication modules over a commutative ring. This topic has been considered by many authors and numerous results have been obtained in this area. After that, Tuganbaev also considered the multiplication module over a noncommutative ring. In this paper, we continue to consider the automorphism-invariance of multiplication modules over a noncommutative ring. We prove that if $R$ is a right duo ring and $M$ is a multiplication, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M$ is projective. Moreover, if $R$ is a right duo, left quasi-duo, CMI ring and $M$ is a multiplication, non-singular, automorphism-invariant, finitely generated right $R$-module with a generating set $\{m_1, \dots , m_n\}$ such that $r(m_i) = 0$ and $[m_iR: M] \subseteq C(R)$ the center of $R$, then $M_R \cong R$ is injective.

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1992.
  • N. V. Dung, D.V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman Scientific & Technical, 1994.
  • Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755-779.
  • P. A. Guil Asensio and A. K. Srivastava, Automorphism-invariant modules satisfy the exchange property, J. Algebra, 388 (2013), 101-106.
  • P. A. Guil Asensio and A. K. Srivastava, Automorphism-invariant modules, in: Noncommutative Rings and Their Applications, Contemp. Math., vol. 634 (2015), 19-30.
  • P. A. Guil Asensio, D. K. Tutuncu and A. K. Srivastava, Modules invariant under automorphisms of their covers and envelopes, Israel J. Math., 206 (2015), 457-482.
  • R. E. Johnson and E. T. Wong, Quasi-injective modules and irreducible rings, J. London Math. Soc., 36 (1961), 260-268.
  • S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, Cambridge University Press, Cambridge, 1990.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  • S. Singh and Y. Al-Shania , Quasi-injective multiplication modules, Comm. Algebra, 28(7) (2000), 3329-3334.
  • S. Singh and A. K. Srivastava, Rings of invariant module type and automorphism-invariant modules, in: Ring Theory and Its Applications , Contemp. Math., vol. 609 (2014), 299-311.
  • P. F. Smith, Some remarks on multiplication modules, Arch. Math. (Basel), 50(3) (1988), 223-235.
  • P. F. Smith, Multiplication modules and projective modules, Period. Math. Hungar., 29(2) (1994), 163-168.
  • A. K. Srivastava, A. A. Tuganbaev and P. A. Guil Asensio, Invariance of Modules Under Automorphisms of Their Envelopes and Covers, Cambridge University Press, Cambridge, 2021.
  • B. Stenstrom, Rings of Quotients, Springer-Verlag, 1975.
  • H. Tachikawa, On left QF-3 rings, Paci c J. Math., 32 (1970), 255-268.
  • L. V. Thuyet, P. Dan and and T. C. Quynh, Modules which are invariant under idempotents of their envelopes, Colloq. Math., 143 (2016), 237-250.
  • A. A. Tuganbaev, Multiplication modules over noncommutative rings, Sb. Math., 194(11-12) (2003), 1837-1864.
  • A. A. Tuganbaev, Multiplication modules, J. Math. Sci. (N.Y.), 123(2) (2004), 3839-3905.
  • A. A. Tuganbaev, Multiplication modules and ideals, J. Math. Sci. (N.Y.), 136(4) (2006), 4116-4130.
  • A. A. Tuganbaev, Automorphism-invariant modules, J. Math. Sci. (N.Y.), 206 (2015), 694-698.
  • A. A. Tuganbaev, Automorphism-invariant non-singular rings and modules, J. Algebra, 485 (2017), 247-253.
  • R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • H. P. Yu, On quasi-duo rings, Glasgow Math. J., 37 (1995), 21-31.
There are 24 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Le Van Thuyet

Truong Cong Quynh

Early Pub Date December 28, 2023
Publication Date July 12, 2024
Published in Issue Year 2024 Volume: 36 Issue: 36

Cite

APA Thuyet, L. V., & Quynh, T. C. (2024). On Automorphism-invariant multiplication modules over a noncommutative ring. International Electronic Journal of Algebra, 36(36), 73-88. https://doi.org/10.24330/ieja.1411145
AMA Thuyet LV, Quynh TC. On Automorphism-invariant multiplication modules over a noncommutative ring. IEJA. July 2024;36(36):73-88. doi:10.24330/ieja.1411145
Chicago Thuyet, Le Van, and Truong Cong Quynh. “On Automorphism-Invariant Multiplication Modules over a Noncommutative Ring”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 73-88. https://doi.org/10.24330/ieja.1411145.
EndNote Thuyet LV, Quynh TC (July 1, 2024) On Automorphism-invariant multiplication modules over a noncommutative ring. International Electronic Journal of Algebra 36 36 73–88.
IEEE L. V. Thuyet and T. C. Quynh, “On Automorphism-invariant multiplication modules over a noncommutative ring”, IEJA, vol. 36, no. 36, pp. 73–88, 2024, doi: 10.24330/ieja.1411145.
ISNAD Thuyet, Le Van - Quynh, Truong Cong. “On Automorphism-Invariant Multiplication Modules over a Noncommutative Ring”. International Electronic Journal of Algebra 36/36 (July 2024), 73-88. https://doi.org/10.24330/ieja.1411145.
JAMA Thuyet LV, Quynh TC. On Automorphism-invariant multiplication modules over a noncommutative ring. IEJA. 2024;36:73–88.
MLA Thuyet, Le Van and Truong Cong Quynh. “On Automorphism-Invariant Multiplication Modules over a Noncommutative Ring”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 73-88, doi:10.24330/ieja.1411145.
Vancouver Thuyet LV, Quynh TC. On Automorphism-invariant multiplication modules over a noncommutative ring. IEJA. 2024;36(36):73-88.