An alternative perspective on injectivity of modules

Given modules M and N, M is said to be N-subinjective if for every extension K of N and every homomorphism φ:N→M there exists a homomorphism ϕ:K→M such that ϕ|N=φ. For a module M, the subinjectivity domain of M is defined to be the collection of all modules N such that M is N-subinjective. As an opposite to injectivity, a module M is said to be indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the injective modules. Properties of subinjectivity domains and of indigent modules are studied. In particular, the existence of indigent modules is determined for some families of rings including the ring of integers and Artinian serial rings. It is also shown that some rings (e.g. Artinian chain rings) have no middle class in the sense that all modules are either injective or indigent. For various classes of modules (such as semisimple, singular and projective), necessary and sufficient conditions for the existence of indigent modules of those types are studied. Indigent modules are analog to the so-called poor modules, an opposite of injectivity (in terms of injectivity domains) recently studied in papers by Alahmadi, Alkan and López-Permouth and by Er, López-Permouth and Sökmez. Relations between poor and indigent modules are also investigated here.