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Quasi-dual Baer modules. (English) Zbl 1483.16011

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.

MSC:

16D80 Other classes of modules and ideals in associative algebras
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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
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