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Generalizations of \(ss\)-supplemented modules. (English) Zbl 1496.16001

Summary: We introduce the concept of (strongly) \(ss\)-radical supplemented modules. We prove that if a submodule \(N\) of \(M\) is strongly \(ss\)-radical supplemented and \(Rad(M/N)=M/N\), then \(M\) is strongly \(ss\)-radical supplemented. For a left good ring \(R\), we show that \(Rad(R)\subseteq Soc(_RR)\) if and only if every left \(R\)-module is \(ss\)-radical supplemented. We characterize the rings over which all modules are strongly \(ss\)-radical supplemented. We also prove that over a left \(WV\)-ring every supplemented module is \(ss\)-supplemented.

MSC:

16D10 General module theory in associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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References:

[1] B.N. Türkmen, E. Türkmen, A note on a generalization of injective modules , Carpathian Mathematical Publications: Vol. 12 No. 2 (2020) · Zbl 1466.16002
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