Tribak, Rachid; Talebi, Yahya; Hosseinpour, Mehrab Quasi-dual Baer modules. (English) Zbl 1483.16011 Arab. J. Math. 10, No. 2, 497-504 (2021). Summary: Let \(R\) be a ring and let \(M\) be an \(R\)-module with \(S=\mathrm{End}_R(M)\). The module \(M\) is called quasi-dual Baer if for every fully invariant submodule \(N\) of \(M\), \(\{\phi\in S\mid \mathrm{Im}\phi\subseteq N\}= eS\) for some idempotent \(e\) in \(S\). We show that \(M\) is quasi-dual Baer if and only if \(\sum_{\varphi\in\mathfrak{I}}\varphi (M)\) is a direct summand of \(M\) for every left ideal \(\mathfrak{I}\) of \(S\). The \(R\)-module \(R_R\) is quasi-dual Baer if and only if \(R\) is a finite product of simple rings. Other characterizations of quasi-dual Baer modules are obtained. Examples which delineate the concepts and results are provided. Cited in 1 Document MSC: 16D80 Other classes of modules and ideals in associative algebras Keywords:quasi-dual Baer modules PDFBibTeX XMLCite \textit{R. Tribak} et al., Arab. J. Math. 10, No. 2, 497--504 (2021; Zbl 1483.16011) Full Text: DOI References: [1] Amouzegar, T.; Talebi, Y., On Quasi-dual Baer modules, TWMS J. Pure Appl. Math., 4, 1, 78-86 (2013) · Zbl 1297.16010 [2] Anderson, FW; Fuller, KR, Rings and Categories of Modules (1974), New York: Springer, New York · Zbl 0301.16001 · doi:10.1007/978-1-4684-9913-1 [3] Goodearl, KR, Von Neumann Regular Rings (1979), London: Pitman, London · Zbl 0411.16007 [4] Keskin Tütüncü, D.; Tribak, R., On dual Baer modules, Glasg. Math. J., 52, 2, 261-269 (2010) · Zbl 1215.16007 · doi:10.1017/S0017089509990334 [5] Lam, TY, A First Course in Noncommutative Rings, Second edition, Graduate Texts in Mathematics 131 (2001), New York: Springer, New York · Zbl 0980.16001 · doi:10.1007/978-1-4419-8616-0 [6] Lee, G.; Tariq Rizvi, S., Endoprime modules and their direct sums, J. Algebra Appl., 17, 8, 1850155 (2018) · Zbl 1414.16005 · doi:10.1142/S0219498818501554 [7] Lomp, C., A counterexample for problem on quasi-Baer modules, Taiwanese J. Math., 21, 6, 1277-1281 (2017) · Zbl 1386.16002 · doi:10.11650/tjm/8028 [8] Sharpe, DW; Vamos, P., Injective Modules (1972), Cambridge: Cambridge University Press, Cambridge · Zbl 0245.13001 [9] Tribak, R., On weak dual Rickart modules and dual Baer modules, Commun. Algebra, 43, 8, 3190-3206 (2015) · Zbl 1331.16006 · doi:10.1080/00927872.2014.913056 [10] Wisbauer, R., Foundations of Module and Ring Theory (1991), Philadelphia: Gordon and Breach, Philadelphia · Zbl 0746.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.