Characterization of rings using direct-projective modules and direct- injective modules (Q1802154)

A module \(M\) is said to be direct-projective if, given any summand \(N\) of \(M\) with projection \(p:M \to N\) and any epimorphism \(f: M\to N\), there is a \(g\in \text{End}_ R(M)\) such that \(gf = p\). Similarly, a module \(M\) is called direct-injective if given any summand \(N\) of \(M\) with injection \(i: N\to M\) and any monomorphism \(f: N\to M\), there exists a \(g\in \text{End}_ R(M)\) such that \(fg = i\). The author uses direct-projective and direct-injective modules to give new characterizations of hereditary, semihereditary and semisimple rings. For example, he shows that the following are equivalent for any ring \(R\): (1) \(R\) is left hereditary. (2) Every submodule of a projective \(R\)-module is direct-projective. (3) Every principal left ideal of \(S = \text{End}_ R(F)\) is direct- projective for any free \(R\)-module \(F\). (4) Every factor module of an injective \(R\)-module is direct-injective. (5) Every sum of two injective submodules of an \(R\)-module is direct-injective. (6) Every sum of two isomorphic injective submodules of an \(R\)-module is direct-injective.