On strongly almost hereditary rings (Q1861072)

Let \(R\) be a basic Artinian ring with identity, and let \(M\) and \(N\) be \(R\)-modules. Then \(M\) is called almost \(N\)-projective (almost \(N\)-injective) if for any homomorphism \(f\colon M\to L\) (\(f'\colon L\to M\)) and any epimorphism \(p\colon N\to L\) (monomorphism \(p'\colon L\to N\)) either there exists a homomorphism \(h\colon M\to N\) (\(h'\colon N\to M\)) such that \(f=ph\) (\(f'=h'p'\)) or there exists a nonzero direct summand \(N'\) of \(N\) and a homomorphism \(t\colon N'\to M\) (\(t'\colon M\to N'\)) such that \(ft=p\lambda\) (\(t'f'=\pi p'\)), where \(\lambda\colon N'\to N\) is inclusion and \(\pi\colon N\to N'\) is projection. Further, \(M\) is called almost projective (almost injective) if \(M\) is almost \(N\)-projective (almost \(N\)-injective) for each finitely generated \(R\)-module \(N\). Finally, \(R\) is a right strongly almost hereditary (SAH) ring if every submodule of a finitely generated projective right \(R\)-module is almost projective. Then \(R\) is a right SAH ring if and only if every factor module of an injective left \(R\)-module is a direct sum of an injective module and a finitely generated almost injective module. An example is given of a right SAH ring that is not a left SAH ring. The following statements are equivalent: (1) the Jacobson radical of each finitely generated almost projective right \(R\)-module is projective; (2) every submodule of a finitely generated almost projective right \(R\)-module is almost projective; (3) for any injective or finitely generated almost injective left \(R\)-module \(M\), \(M/\text{Soc}(M)\) is a direct sum of an injective and finitely generated almost injective modules; and (4) every factor module of an injective or finitely generated almost injective right \(R\)-module is a direct sum of an injective module and finitely generated almost injective modules.