A mathematical preprint sets out several ways to characterize hereditary QF-3+ rings. Its central test compares two technical classes of modules, stable modules and projective modules: they form a torsion theory, the paper’s formal pairing of the two classes, exactly when the ring is hereditary QF-3+.
The result gives ring theorists a common thread through constructions that can otherwise look separate. The paper links torsion theories and radicals to module categories, then carries the analysis into injective envelopes and the maximal left ring of quotients.
The first test is a relationship between modules
This is a formal mathematical analysis. It works with associative rings with identity and unital left modules, and it uses radicals, pre-torsion theories and torsion theories as its main framework.
For a left hereditary ring, the paper gives an additional route to the same characterization. It uses a reflector onto projective modules that preserves monomorphisms, meaning that an injective map remains a monomorphism after the reflection. The equivalent module-level condition is that stable modules contain no nonzero projective submodules.
That pairing moves QF-3+ from a label attached to a ring to a relationship visible in its modules. The result is an if-and-only-if statement, so either side can serve as a characterization under the paper’s hereditary assumption.
The stable side holds together
The stable side is not treated as a loose collection of examples. For a hereditary QF-3+ ring, stable modules form an Abelian subcategory, and that subcategory is closed under essential extensions and injective envelopes. The stable category itself has injective envelopes as well.
A related theorem follows the largest stable-submodule radical. The radical commutes with taking injective envelopes, and applying it to an injective module produces another injective module. The operation therefore remains compatible with the envelope construction.
Together, these results give the stable modules an internal structure that can be handled on its own terms. Closure, injective envelopes and the radical remain tied to the same stable framework.
Projectives and injectives meet
The paper draws a sharp line around the projective category. The reflector from all left modules to projective modules is left exact exactly in the semisimple case, and the category of projective modules is Abelian exactly in that same case. Semisimplicity is the dividing condition in both statements.
Projective-injective modules receive a parallel treatment. They form reflective Abelian subcategories both inside the projective modules and inside all modules, while the reflector from all modules to this projective-injective subcategory is exact.
The construction also has a functorial form. Injective envelopes of projective modules extend uniquely to an additive functor whose values lie among projective-injective modules, with the embeddings natural.
A quotient ring gives another route
One of the paper’s most striking moves is to put a ring structure on the injective envelope of the regular module. That structure is unique, while still extending the envelope’s given left-module structure.
The constructed ring is isomorphic to Qlmax(Λ), the maximal left ring of quotients. This identifies two descriptions of the same object up to isomorphism: one begins with the regular module and its injective envelope, while the other uses the maximal quotient-ring construction.
For a hereditary QF-3+ ring, the maximal left ring of quotients is semisimple and left Artinian. The paper also states a bimorphism from the ring to a semisimple left Artinian ring, adding a map-based formulation to the quotient-ring result.
That quotient supplies another if-and-only-if test. For a left hereditary ring, QF-3+ is equivalent to Qlmax(Λ) being semisimple and projective as a left module.
The paper gives a second version that looks from the right as well. It characterizes QF-3+ by requiring Qlmax(Λ) to be projective as a right Λ-module and every left Qlmax(Λ)-module to be projective over Λ.
The result stays within a formal setting
Seen together, the findings form a chain rather than a single isolated theorem. Stable and projective modules supply the torsion-theoretic test; the categorical results describe how those classes behave; injective envelopes and the maximal quotient ring provide further structural translations.
These are conditional results, not blanket claims about every ring. The stable-module statements are made for hereditary QF-3+ rings, while the reflector and maximal-quotient criteria are stated with left-hereditary assumptions where specified.
The author states that right-sided versions of all the preceding statements are valid. The supplied document is an arXiv version-one preprint dated 25 August 2026.
Paper data and sources
Original title: Hereditary QF-$3^{+}$ rings
Authors: Dali Zangurashvili
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text