Projection invariant $t$-Baer and related modules

  • Yeliz Kara Department of Mathematics, Bursa Uludağ University, Görükle, Turkey
Keywords: t-essential, t-extending, projection invariant submodule, pi-e.Baer module, t-Baer, endomorphism ring


UDC 512.5

We investigate the concepts of projection invariant $t$-extending modules and projection invariant $t$-Baer modules, which are generalized to those $\pi$-extending and $t$-Baer notions, respectively. Several structural properties are obtained and some applications are developed. It is proved that the $\pi$-$t$-extending modules and $\pi$-$t$-e. Baer modules are connected with each other. Moreover, we obtain a characterization for $\pi$-$t$-extending modules relative to the annihilator conditions.


F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer-Verlag, New York (1992).

S. Asgari, A. Haghany, Generalizations of $t$-extending modules relative to fully invariant submodules, J. Korean Math. Soc., 49, № 3, 503–514 (2012).

S. Asgari, A. Haghany, A. R. Rezaei, Modules whose $t$-closed submodules have a summand as a complement, Comm. Algebra, 42, № 12, 5299–5318 (2014).

S. Asgari, A. Haghany, t-Extending modules and $t$-Baer modules, Comm. Algebra, 39, № 5, 1605–1623 (2011).

G. F. Birkenmeier, J. Y. Kim, J. K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J., 40, 247–253 (2000).

G. F. Birkenmeier, Y. Kara, A. Tercan, $pi$-Baer rings, J. Algebra and Appl., 17, № 2, Article 1850029 (2018).

G. F. Birkenmeier, Y. Kara, A. Tercan, $pi$-Endo Baer modules, Comm. Algebra, 48, № 3, 1132–1149 (2020).

G. F. Birkenmeier, B. J. Müller, S. T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra, 30, № 3, 1395–1415 (2002).

G. F. Birkenmeier, J. K. Park, S. T. Rizvi, Extensions of rings and modules, Birkhäuser, New York (2013).

G. F. Birkenmeier, A. Tercan, C. C. Yücel, The extending condition relative to sets of submodules, Comm. Algebra, 42, 764–778 (2014).

G. F. Birkenmeier, A. Tercan, C. C. Yücel, Projection invariant extending rings, J. Algebra and Appl., 115, Article~1650121 (2016).

A. W. Chatters, S. M. Khuri, Endomorphism rings of modules over non-singular CS rings, J. London Math. Soc., 21, 434–444 (1980).

W. E. Clark, Baer rings which arise from certain transitive graphs, Duke Math. J., 33, 647–656 (1966).

L. Fuchs, Infinite Abelian groups, vol. II, Acad. Press, New York, London (1973).

K. R. Goodearl, Nonsingular rings and modules, Marcel Dekker, New York (1976).

Y. Kara, A. Tercan, Modules whose certain submodules are essentially embedded in direct summands, Rocky Mountain J. Math., 46, № 2, 519–532 (2016).

Y. Kara, A. Tercan, On the inheritance of the strongly $pi$-extending property, Comm. Algebra, 45, № 8, 3627–3635 (2016).

I. Kaplansky, Rings of operators, Benjamin, New York (1968).

T. Y. Lam, Lectures on modules and rings, Springer, Berlin (1999).

S. H. Mohamed, B. J. Muller, Continuous and discrete modules, London Math. Soc. Lecture Note Ser., 147, Cambridge Univ. Press, Cambridge (1990).

S. T. Rizvi, C. S. Roman, Baer and quasi-Baer modules, Comm. Algebra, 32, 103–123 (2004).

R. Wisbauer, M. F. Yousif, Y. Zhou, Ikeda–Nakayama modules, Beitr. Algebra und Geom., 43, № 1, 111–119 (2002).

How to Cite
Kara, Y. “Projection Invariant $t$-Baer and Related Modules”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 10, Oct. 2023, pp. 1354- 1365, doi:10.3842/umzh.v75i10.7244.
Research articles