On lifting modules relative to the class of all singular modules. (Q763693)

In this paper, the author generalizes the notions of lifting module, amply supplemented module and weakly supplemented module, respectively, to \(\delta\)-lifting module, amply \(\delta\)-supplemented module and weakly \(\delta\)-supplemented module using the notion of \(\delta\)-small submodule introduced by Y. Q. Zhou. After studying the basic properties of amply \(\delta\)-supplemented modules in Section 2, in Section 3 he studies \(\delta\)-coclosed submodules. In the main Section 4, the author studies \(\delta\)-lifting modules. Finally, in Section 5, the author generalizes the well-known result of Bass characterizing (semi-) perfect rings in terms of lifting modules to \(\delta\)- (semi-) perfect rings. I add few of my observations. (i) In \((1)\Rightarrow(2)\) of Theorem 4.4, the author seems to prove that every \(\delta\)-coclosed submodule \(K\) of a \(\delta\)-lifting module \(M\) is a direct summand of \(M\). In this proof, why \(K/H\) is singular is not clear (here the author doesn't seem to use any additional hypothesis). (ii) In the proof of \((1)\Rightarrow(2)\) of Proposition 4.8 it needs to be proved that \(M/N\cap X\) is singular which doesn't seem to be obvious. (iii) If the proof of \((1)\Rightarrow(2)\) of Theorem 4.4 is not rectified, in the proof of Lemma 4.9 it is necessary that \(M/N'=(M_1+N')/ N'\oplus(M_2+N')/ N'\). The same observation regarding the proof of (3) implies \(M\) is \(\delta\)-lifting in Theorem 4.11.