Characterization of rings using weakly projective modules. II (Q1297386)

[For part I cf. the first author, ibid. 25, No. 2, 91-98 (1997; Zbl 0896.16005).] \textit{H. Zöschinger} [Arch. Math. 25, 241-253 (1974; Zbl 0306.16022)] called a module \(M\) weakly projective if, for each pair \((A,B)\) of submodules of \(M\) with \(M=A+B\), there exists an endomorphism \(f\colon M\to M\) such that \(\text{Im}(f)\subseteq A\) and \(\text{Im}(1-f)\subseteq B\). The author characterizes left \(pp\)-rings, semisimple rings, semiperfect rings, and semiregular rings using weakly projective modules.