On Automorphism-invariant multiplication modules over a noncommutative ring

Yıl 2024, Cilt: 36 Sayı: 36, 73 - 88, 12.07.2024

Le Van Thuyet Truong Cong Quynh

https://doi.org/10.24330/ieja.1411145

Öz

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.

Anahtar Kelimeler

Automorphism-invariant module, duo ring, quasi-duo ring, multiplication module, commutative multiplication of ideals

Kaynakça

• 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.