Poor Modules: The Opposite of Injectivity

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.

Section: Modules Whose Subinjectivity Domain Consists Of Only Injectimentioning

confidence: 99%

Section: Indigent Modules Of Specific Typesmentioning

confidence: 99%

See 1 more Smart Citation

An alternative perspective on injectivity of modules

Aydoğdu

López-Permouth

2011

“…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).…”

Section: Introductionmentioning

confidence: 99%

Rings and modules characterized by opposites of injectivity

Alizade

Büyükaşık

2014

In a recent paper, Aydoǧdu and López-Permouth have defined a module M to be N -subinjective if every homomorphism N → M extends to some E(N ) → M , where E(N ) is the injective hull of N . Clearly, every module is subinjective relative to any injective module. Their work raises the following question: What is the structure of a ring over which every module is injective or subinjective relative only to the smallest possible family of modules, namely injectives? We show, using a dual opposite injectivity condition, that such a ring R is isomorphic to the direct product of a semisimple Artinian ring and an indecomposable ring which is (i) a hereditary Artinian serial ring with J 2 = 0; or (ii) a QF-ring isomorphic to a matrix ring over a local ring. Each case is viable and, conversely, (i) is sufficient for the said property, and a partial converse is proved for a ring satisfying (ii). Using the above mentioned classification, it is also shown that such rings coincide with the fully saturated rings of Trlifaj except, possibly, when von Neumann regularity is assumed. Furthermore, rings and abelian groups which satisfy these opposite injectivity conditions are characterized.

“…In fact, by [1,Proposition 3.1], semisimple modules are the only ones relative to which every module is injective. In [1], Alahmadi, Alkan and López-Permouth introduced modules M whose domain of injectivity In −1 (M) = {N ∈ Mod − R | M is N-injective} is smallest possible, consisting only of semisimple modules in Mod − R, and called such modules poor. They studied the question of which rings have poor modules and gave it some partial answers.…”

Section: Introductionmentioning

confidence: 99%

Rings whose modules have maximal or minimal injectivity domains

López-Permouth

Sökmez

2011

Self Cite

Communicated by Efim Zelmanov Keywords:Injective module Poor module Injectivity domain V-, QI-, SI-, PCI-, QF-ring In a recent paper, Alahmadi, Alkan and López-Permouth defined a module M to be poor if M is injective relative only to semisimple modules, and a ring to have no right middle class if every right module is poor or injective. We prove that every ring has a poor module, and characterize rings with semisimple poor modules. Next, a ring with no right middle class is proved to be the ring direct sum of a semisimple Artinian ring and a ring T which is either zero or of one of the following types: (i) Morita equivalent to a right PCI-domain, (ii) an indecomposable right SI-ring which is either right Artinian or a right V-ring, and such that soc (T T ) is homogeneous and essential in T T and T has a unique simple singular right module, or (iii) an indecomposable right Artinian ring with homogeneous right socle coinciding with the Jacobson radical and the right singular ideal, and with unique non-injective simple right module. In case (iii) either T T is poor or T is a QF-ring with J (T ) 2 = 0. Converses of these cases are discussed. It is shown, in particular, that a QF-ring R with J (R) 2 = 0 and homogeneous right socle has no middle class.