On subinjectivity domains of pure-injective modules

Abstract

A pure-injective module $M$ is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module $M$ is called subinjective relative to a module $N$ if for every extension $K$ of $N$, every homomorphism $N\to M$ can be extended to a homomorphism $K\to M$. The subinjectivity domain of the module $M$ is defined to be the class of modules $N$ such that $M$ is $N$-subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigent modules are studied. The structure of a ring over which every simple, uniform, or indecomposable pure-injective module is injective or subinjective relative only to the smallest possible family of modules is investigated.