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
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.
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.
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.