Completion of diagrams by automorphisms and Bass' first stable range condition
Weak projectivity and injectivity conditions for a module \(P\) are studied and related to the stable range of the endomorphism ring of \(P\). For instance, \(P\) is defined to be quasi-projective (respectively, epi-projective) if for any epimorphism \(g:P\to M\) and any homomorphism (respectively, epimorphism) \(f:P\to M\), there exists a homomorphism (respectively, epimorphism) \(h:P\to P\) such that \(f=gh\). One of the main theorems of the paper gives a number of equivalent conditions for the endomorphism ring of a quasi-projective module \(P\) to have stable range 1; in particular, this occurs if and only if \(P\) is epi-projective. Another main theorem enlarges the list of equivalent conditions by specializing to free modules of rank 1. For example, a ring \(R\) has stable range 1 if and only if every cyclic right \(R\)-module \(M\) is uniquely generated, meaning that whether \(M= xR=yR\), there exists a unit \(u\in R\) such that \(x=yu\). The paper also contains a number of general results on epi-projectivity and unique generation, as well as a discussion of the dual problems for quasi-injective modules.
scientific article; zbMATH DE number 815921