An alternative perspective on injectivity of modules

Abstract

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.

Rings and modules characterized by opposites of injectivity

“…As a by-product of our results the answer to another question raised in [3] is obtained: There do exist poor modules (i.e. modules injective relative only to semisimples, see [1] and [7]) which are not indigent, in particular over a PCI-domain which is not a division ring (Remark 9).…”

“…So, the condition (i) implies that any such sum is injective, hence R is right Noetherian. Furthermore, R is right hereditary by [3,Proposition 2.9].…”

“…that of a module with respect to a fixed module, inspired by Baer's criterion. Introduced in a similar vein, the kind of injectivity (namely, subinjectivity) and the opposite of injectivity induced by it (namely, indigence), as discussed by Aydoǧdu and López-Permouth in [3], seem to offer a new perspective on the topic and to be abundant with interesting questions: Let R be an associative ring with identity and M and N be unital right R-modules. M is said to be N -subinjective if every homomorphism N → M can be extended to some homomorphism E(N ) → M , where E(N ) is the injective hull of N .…”

“…So, the smallest possible subinjectivity domain is the class of injective modules. A module with such a subinjectivity domain is defined in [3] to be indigent. The existence of indigent modules for an arbitrary ring is unknown, but an affirmative answer is known for some rings such as Z and Artinian serial rings (see [3]).…”

“…In this paper, we address some questions raised by and studied in [3]. The first question considered here is the following: What is the structure of a ring over which every right module is indigent or injective?…”