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Hereditarily and Super Bassian Modules over Certain Rings

This paper characterizes hereditarily and super Bassian modules over non-primitive Dedekind prime rings and Dedekind domains, establishing structural criteria and proving that for singular modules, the super Bassian property is equivalent to the Bassian property, while for arbitrary modules over these rings, being super Bassian implies being hereditarily Bassian.

Original authors: Peter Vassilev Danchev, Truong Cong Quynh, Jan Žemlička

Published 2026-03-26

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CC BY 4.0

Original authors: Peter Vassilev Danchev, Truong Cong Quynh, Jan Žemlička

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect designing a massive, complex city called The Ring. In this city, there are many different types of buildings called Modules. These buildings are made of smaller rooms (submodules) and can be reshaped or broken down into smaller structures (quotients).

For a long time, mathematicians have been studying a special property of these buildings called being "Bassian."

The Core Concept: The "Bassian" Rule

Think of a Bassian building as a structure that is very "rigid" or "honest."

The Rule: If you try to shrink the building by removing a room (a submodule) and somehow the smaller building still looks exactly like the original big building (a mathematical concept called a "monomorphism"), then you must have removed nothing at all.
In plain English: A Bassian building cannot be "shrunk" into a smaller version of itself without losing its identity. If you can shrink it and it still looks the same, the building was actually empty to begin with.

The paper you provided explores two extreme versions of this rule, asking: "What happens if we apply this rule to every part of the building, or every version of the building?"

1. The "Super Bassian" Building (The Ultimate Shrink-Proof)

Imagine a Super Bassian building. This is a building so sturdy that no matter how you slice it, cut it, or reshape it (any "epimorphic image"), the resulting piece is still a Bassian building.

The Analogy: Think of a diamond. If you chip off a piece, the remaining piece is still a diamond. If you cut it in half, both halves are still diamonds. A Super Bassian module is like a diamond that remains "rigid" even after being broken apart.
The Discovery: The authors found that for certain types of cities (specifically "non-primitive Dedekind prime rings"), a building is Super Bassian if and only if:
1. It has a small, manageable "core" (a finitely generated part).
2. The rest of the building is a "ghost" (singular) that doesn't add much structural weight.
3. The "ghost" parts are neatly organized into small, finite groups.

Key Insight: If a building is Super Bassian, it turns out it is also Hereditarily Bassian. It's like saying, "If a building is so tough that every piece of it is tough, then every piece of it is also tough in a specific way."

2. The "Hereditarily Bassian" Building (The Family Tree Rule)

Now, imagine a Hereditarily Bassian building. This is a building where every single room, every corner, and every sub-room is a Bassian building.

The Analogy: Think of a family tree. If a family is "Hereditarily Bassian," it means every ancestor, every parent, every child, and every grandchild follows the "rigid" rule. You can't find a single weak link in the chain.
The Discovery: The authors proved that for these specific cities, a building is Hereditarily Bassian if:
1. The "ghost" parts (singular components) are neatly organized and finite.
2. There is a small, solid core that holds the whole thing together.

The Big Surprise: The Relationship Between the Two

The most interesting part of the paper is the relationship between these two types of buildings in these specific cities:

The Twist: If a building is Super Bassian (tough in every shape), it is automatically Hereditarily Bassian (tough in every part).
The Exception: The reverse isn't always true! You can have a building where every room is tough (Hereditarily Bassian), but if you try to reshape the whole building, it might fall apart (not Super Bassian).

Real-World Example from the Paper:

The Rational Numbers ( $\mathbb{Q}$ ): Think of this as a building where every single room is perfect and rigid. It is Hereditarily Bassian. However, if you try to reshape it, it loses its identity. So, it is NOT Super Bassian.
A Specific Vector Space: Imagine a building made of infinite water. It is Super Bassian (it holds its shape no matter how you pour it), but if you look at a specific pipe inside it, that pipe is weak. So, it is NOT Hereditarily Bassian.

Why Does This Matter?

The authors are essentially creating a blueprint for mathematicians.

Before: We knew what a "Bassian" building looked like.
Now: We know exactly what the "Super" and "Hereditary" versions look like in complex, non-commutative cities (rings that don't follow standard multiplication rules like $A \times B = B \times A$ ).

They have solved a puzzle that connects these rigid structures to the concept of finiteness. Basically, they found that for these buildings to be this "super tough," they must have a finite, manageable core, even if the building itself is huge.

Summary in One Sentence

This paper acts as a guidebook for architects of mathematical structures, proving that in certain complex cities, if a building is unbreakable in every shape (Super Bassian), it is guaranteed to be unbreakable in every part (Hereditarily Bassian), provided the building has a finite, organized core.

1. Problem Statement and Motivation

The paper investigates the structural properties of modules over specific classes of rings, focusing on two strengthened variations of the Bassian property.

Bassian Module: A module $M$ is Bassian if the existence of a monomorphism $M \to M/N$ for a submodule $N$ implies $N=0$ .
Super Bassian Module: A module $M$ is super Bassian if every epimorphic image (factor module) of $M$ is Bassian. (The authors note this is equivalent to the module being Hopfian).
Hereditarily Bassian Module: A module $M$ is hereditarily Bassian if every submodule of $M$ is Bassian.

The Core Problem: While Bassian modules have been studied (e.g., by Rowen, Small, Tuganbaev, and Blacher), the paper seeks to characterize when modules are super Bassian or hereditarily Bassian, specifically over non-primitive Dedekind prime rings (a generalization of Dedekind domains to the non-commutative setting) and semi-local rings with nilpotent Jacobson radicals. The authors aim to determine the structural criteria for these properties and explore the relationship between them.

2. Methodology

The authors employ a combination of homological algebra, module theory, and ring theory techniques:

Structural Decomposition: Utilizing the decomposition of modules into singular and non-singular parts, and analyzing primary components relative to maximal invertible ideals.
Essential Submodules and Injective Envelopes: Using properties of essential submodules ( $B \unlhd M$ ) and injective envelopes ( $E(M)$ ) to derive contradictions regarding Goldie dimensions.
Induction and Transfinite Arguments: Employing induction on socle sequences and transfinite induction to analyze semi-Artinian modules and their lengths.
Reduction to Known Theorems: Leveraging existing results from the literature (specifically from reference [7], likely a foundational text on HNP rings and Goldie dimension) to establish necessary and sufficient conditions.
Counter-Examples: Constructing specific examples (using matrix rings and vector spaces) to demonstrate the independence of the super Bassian and hereditarily Bassian properties.

3. Key Contributions and Results

A. Super Bassian Modules

The paper establishes a complete characterization for modules over non-primitive Dedekind prime rings (HNP rings that are not primitive).

Theorem 1.13: A module $M$ $M$ over a non-primitive Dedekind prime ring is super Bassian if and only if:
1. There exists a finitely generated submodule $H$ of $M$ such that the quotient $M/H$ is singular.
2. All primary components of $M/H$ are finitely generated (which, over these rings, implies they are Noetherian).
Corollary 1.14: For a singular module $M$ over such a ring, the properties of being Bassian, super Bassian, and having Noetherian primary components are all equivalent.
Corollary 1.15 (Commutative Case): Over a Dedekind domain, $M$ is super Bassian iff there exists a finitely generated torsion-free submodule $F$ such that $M/F$ is torsion and all primary components of $M/F$ are finitely generated.
General Rings:
- Over right Artinian rings, a module is super Bassian iff it is finitely generated (Proposition 1.6).
- Over right Noetherian rings, any finitely generated module is super Bassian (Lemma 1.7).

B. Hereditarily Bassian Modules

The authors provide criteria for when every submodule is Bassian.

Theorem 2.6 (Semi-local Rings): Let $R$ $R$ be a semi-local ring with a nilpotent Jacobson radical. A module $M$ $M$ is hereditarily Bassian if and only if its socle, $\text{Soc}(M)$ $Soc (M)$ , is finitely generated.
- Significance: This generalizes previous results for quasi-Frobenius rings and shows that finite generation of the module itself is not required (Example 2.7 provides an infinitely generated hereditarily Bassian module over a right Artinian ring).
Theorem 2.9 (Non-primitive Dedekind Prime Rings): A module $M$ $M$ is hereditarily Bassian iff:
1. Each primary component of the singular submodule $\text{Sing}(M)$ is finitely generated (Noetherian).
2. There exists a finitely generated submodule $G$ such that $M/G$ is singular.
Corollary 2.11 (Commutative Case): Over a Dedekind domain, $M$ is hereditarily Bassian iff all primary components of $M$ are finitely generated and there exists a finitely generated torsion-free submodule $F$ such that $M/F$ is torsion.

C. Relationship Between the Properties

A significant finding is the asymmetry between the two properties over non-primitive Dedekind prime rings:

Corollary 2.10(1): If a module $M$ is super Bassian, it is necessarily hereditarily Bassian.
Independence: The converse is not true.
- Example 2.12: The rational group $\mathbb{Q}$ is hereditarily Bassian but not super Bassian.
- Example 2.13: A specific local perfect ring $R$ is super Bassian (as a module over itself) but not hereditarily Bassian because its Jacobson radical is not Bassian.

4. Significance and Impact

Unification of Concepts: The paper successfully unifies the study of Bassian, super Bassian, and hereditarily Bassian modules, clarifying their interdependencies in the context of Dedekind prime rings.
Extension of Literature: It extends recent work by Tuganbaev and Blacher (2026) from Abelian groups and specific algebras to the broader and more complex setting of non-commutative Dedekind prime rings.
Structural Insight: The results provide precise structural "checklists" (e.g., finiteness of socles, properties of singular quotients) for researchers to determine if a module possesses these strong Bassian properties without checking every submodule or quotient individually.
Future Directions: The paper concludes by posing open problems regarding the structure of these modules over larger classes of non-commutative rings and the definition of "hereditary" and "super" Bassian rings, inviting further research.

In summary, the paper provides a definitive structural characterization of super and hereditarily Bassian modules over Dedekind prime rings, proving that while super Bassian implies hereditarily Bassian in this context, the reverse is false, and offering concrete criteria based on the finiteness of singular quotients and primary components.