Characterization of rings using weakly projective modules (Q1384882)

A module \(M\) is weakly projective if for every pair \((A,B)\) of submodules of \(M\) with \(M=A+B\), there exists an endomorphism \(f\colon M\to M\) with \(\text{Im}(f)\subseteq A\) and \(\text{Im}(1-f)\subseteq B\). The author gives a number of characterizations of rings which generalize well-known results of \textit{J. S. Golan} [Isr. J. Math. 8, 34-38 (1970; Zbl 0199.35502)]\ and other authors, of which the following is a sample result. A ring \(R\) is left-hereditary iff every submodule of a projective \(R\)-module is weakly projective.