Free modules over Hjelmslev rings in which not every maximal linearly independent subset is a basis
It is proved that for an \(H\)-ring \(R\), every maximal linearly independent subset of a free left \(R\)-module \(V\) is a basis of \(V\) if and only if \(R\) is left-noetherian or \(V\) is finitely generated. As a consequence the incidence structure \((P,{\mathcal L})\) with \(\rho=\rho(V^*)\) and \({\mathcal L}=\{\rho(U\backslash Q)| U\) is a free submodule of \(V\) with \(\dim U=2\}\), where \(R\) is an \(H\)-ring with maximal ideal \(N\) and \(V\) an \(R\)- module with torsion \(Q\), \(R^*=R\backslash N\), \(V^*=V\backslash Q\) and \(\rho:V^*\to V^*/R^*:x\mapsto R^*x\) the canonical epimorphism, is a projective Hjelmslev space iff \(R\) is left-noetherian or \(V\) is finitely generated.
scientific article; zbMATH DE number 98732