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Keywords:
fully invariant submodule; quasi-dual Baer module; quasi-dual Baer ring; quasi-t-dual Baer module; quasi-t-dual Baer ring
fully invariant submodule; quasi-dual Baer module; quasi-dual Baer ring; quasi-t-dual Baer module; quasi-t-dual Baer ring
Summary:
Let $R$ be a ring and let $M$ be an $R$-module with $S=\rm{End}_R(M)$. Consider the preradical ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}$ for the category of right $R$-modules Mod-$R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}(M) = \bigcap \{U\leq M\colon M/U$ is small in its injective hull$\}$. The module $M$ is called quasi-t-dual Baer if $\sum_{\varphi \in \mathfrak{I}} \varphi({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M))$ is a direct summand of $M$ for every two-sided ideal $\mathfrak{I}$ of $S$, where ${{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) = {{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}} ({{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}(M))$. In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${{\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}}^2(M) $ is a direct summand of $M$ and ${\mkern 1.5mu\overline{\mkern-3.5mu Z \mkern-.5mu}\mkern 1.5mu}^2(M)$ is a quasi-dual Baer module. It is also shown that any direct summand of a quasi-t-dual Baer module inherits the property. The last part of the paper is devoted to the comparison of the notions of quasi-dual Baer modules and quasi-t-dual Baer modules. Also, right quasi-t-dual Baer rings are investigated.
References:
[1] Amouzegar T., Talebi Y.: On quasi-dual Baer modules. TWMS J. Pure Appl. Math. 4 (2013), no. 1, 78–86. MR 3097682
[2] Amouzegar T., Tütüncü D. K., Talebi Y.: t-dual Baer modules and t-lifting modules. Vietnam J. Math. 42 (2014), no. 2, 159–169. DOI 10.1007/s10013-013-0045-z | MR 3218852
[3] Atani S. E., Khoramdel M., Hesari S. D. P.: T-dual Rickart modules. Bull. Iranian Math. Soc. 42 (2016), no. 3, 627–642. MR 3518208
[4] Clark J., Lomp C., Vanaja N., Wisbauer R.: Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Birkhäuser, Basel, 2006. MR 2253001 | Zbl 1102.16001
[5] Clark W. E.: Twisted matrix units semigroup algebras. Duke Math. J. 34 (1967), 417–424. DOI 10.1215/S0012-7094-67-03446-1 | MR 0214626 | Zbl 0204.04502
[6] Cozzens J. H.: Homological properties of the ring of differential polynomials. Bull. Amer. Math. Soc. 76 (1970), 75–79. DOI 10.1090/S0002-9904-1970-12370-9 | MR 0258886
[7] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R.: Extending Modules. Pitman Research Notes in Mathematics Series, 313, Longman Scientific & Technical, Harlow, 1994. MR 1312366 | Zbl 0841.16001
[8] Haghany A., Karamzadeh O. A. S., Vedadi M. R.: Rings with all finitely generated modules retractable. Bull. Iranian Math. Soc. 35 (2009), no. 2, 37–45, 270. MR 2642924
[9] Harada M.: On small submodules in the total quotient ring of a commutative ring. Rev. Un. Mat. Argentina 28 (1977), 99–102. MR 0469899
[10] Kaplansky I.: Modules over Dedekind rings and valuation rings. Trans. Amer. Math. Soc. 72 (1952), 327–340. DOI 10.1090/S0002-9947-1952-0046349-0 | MR 0046349
[11] Keskin Tütüncü D., Orhan Ertaş N., Smith P. F., Tribak R.: Some rings for which the cosingular submodule of every module is a direct summand. Turkish J. Math. 38 (2014), no. 4, 649–657. DOI 10.3906/mat-1210-15 | MR 3195734
[12] Lee G., Rizvi S. T., Roman C. S.: Dual Rickart modules. Comm. Algebra 39 (2011), no. 11, 4036–4058. DOI 10.1080/00927872.2010.515639 | MR 2855110
[13] Mohamed S. H., Müller B. J.: Continuous and Discrete Modules. London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990. MR 1084376 | Zbl 0701.16001
[14] Rizvi S. T., Roman C. S.: Baer and quasi-Baer modules. Comm. Algebra 32 (2004), no. 1, 103–123. DOI 10.1081/AGB-120027854 | MR 2036224
[15] Smith P. F.: Modules with many homomorphisms. J. Pure Appl. Algebra 197 (2005), no. 1–3, 305–321. DOI 10.1016/j.jpaa.2004.09.001 | MR 2123991
[16] Talebi Y., Vanaja N.: The torsion theory cogenerated by $M$-small modules. Comm. Algebra 30 (2002), no. 3, 1449–1460. DOI 10.1080/00927870209342390 | MR 1892609
[17] Tribak R., Talebi Y., Hosseinpour M.: Quasi-dual Baer modules. Arab. J. Math. (Springer) 10 (2021), no. 2, 497–504. DOI 10.1007/s40065-021-00316-2 | MR 4283974
[18] Tribak R., Talebi Y., Hosseinpour M., Abdi M.: Some results on t-lifting modules. Vietnam J. Math. 46 (2018), no. 3, 653–664. DOI 10.1007/s10013-018-0270-6 | MR 3820454
[19] Tribak R., Talebi Y., Hosseinpour M., Abdi M.: On FI-t-lifting modules. Bol. Soc. Mat. Mex. 26 (2020), no. 3, 973–989. DOI 10.1007/s40590-020-00301-3 | MR 4155340
[20] Tribak R., Tütüncü K. D.: On $\overline{Z}_M$-semiperfect modules. East-West J. Math. 8 (2006), no. 2, 195–205. MR 2442425
[21] Zöschinger H.: Schwach-injective moduln. Period. Math. Hungar. 52 (2006), no. 2, 105–128 (German, English summary). DOI 10.1007/s10998-006-0007-2 | MR 2265652
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