Hereditary rings and relative projectives (Q1197473)

\(R\) is a (right) hereditary ring if and only if every submodule of a projective right \(R\)-module is projective if and only if every factor module of an injective right \(R\)-module is injective. In this paper the author proves analogous results using relative injective and relative projective modules. For example, if \(R\) is a (right) artinian ring, he shows that each of the following is equivalent to \(R\) being a (right) hereditary ring whose Jacobson radical \(J\) is such that \(J^ 2=0\): 1. If \(M\) and \(N\) are local modules and \(M\) is \(N\)-projective, then every submodule of \(M\) is \(N\)-projective. 2. If \(M\) and \(N\) are finitely generated modules and \(M\) is \(N\)- projective, then every submodule of \(M\) is \(N\)-projective. 3. If \(M\) and \(N\) are uniform modules and \(M\) is \(N\)-injective, then every factor module of \(M\) is \(N\)-injective. 4. If \(M\) and \(N\) are finitely generated modules and \(M\) is \(N\)- injective, then every factor module of \(M\) is \(N\)-injective. Several other characterizations of (right) artinian hereditary rings whose Jacobson radical is square zero are also given in terms of relative injective and relative projective modules. Conditions similar to those given in 1. through 4. are also investigated where \(N\)-projective (\(N\)- injective) is replaced by almost \(N\)-projective (almost \(N\)-injective). Among other results, conditions such as these are shown to characterize (two sided) artinian, right almost hereditary rings whose Jacobson radical is square zero.