Singular Generalized Bassian Modules

• Singular generalized Bassian modules are right A–modules defined by the property that an injective homomorphism from M to M/N forces N to be a direct summand.
• They are characterized by injectivity, uniserial submodule structure, and a precise P–primary decomposition that parallels classical Prüfer group behavior.
• Their study over non-primitive Dedekind prime rings provides clear classification and structural insights, linking ideal factorization and uniform dimension.

A singular generalized Bassian module is a right module $M$ over a ring $A$ such that the existence of an injective homomorphism $M\to M/N$ for some submodule $N$ of $M$ implies that $N$ is a direct summand of $M$. The classification and structure of these modules is particularly tractable over non-primitive Dedekind prime rings, a distinguished class of hereditary noetherian prime rings lacking faithful simple modules. The interplay of module-theoretic properties (singularity, injectivity, uniseriality) and ring-theoretic invariants (invertible ideals, Goldie dimension, lattice rigidity) yields a highly structured module category with numerous noncommutative analogues of classical abelian group phenomena (Tuganbaev, 22 Jan 2026).

1. Non-Primitive Dedekind Prime Rings: Definitions and Structure

A ring $A$ is a Dedekind prime ring if it is noetherian and prime, and its classical two-sided ring of fractions $Q$ is semisimple artinian, specifically $Q \cong M_n(D)$ for some division ring $D$, with every nonzero (two-sided) ideal of $A$ invertible in $Q$. $A$ is non-primitive if it admits no faithful simple right module. Such rings satisfy hereditary, noetherian, and prime conditions, and their nonzero two-sided ideals are invertible $A$–$A$-bimodules.

Key features include:

• Every essential one-sided ideal contains a nonzero two-sided ideal.
• The set $\mathcal{P}(A)$ of maximal invertible ideals serves as “prime ideals,” with every nonzero ideal factoring uniquely as a product of elements of $\mathcal{P}(A)$.
• $A$ is a right and left Goldie ring; its uniform dimension equals the matrix size $n$ in $Q$.
• All nonzero (one-sided) ideals are projective and exhibit constant rank one over $Q$.
• Non-primitive Dedekind prime rings, as per the Lenagan–Robson theorem, are the non-artinian bounded hereditary noetherian prime (HNP) rings.

2. Examples and Constructions

Non-primitive Dedekind prime rings encompass many classical and noncommutative contexts:

• Any commutative Dedekind domain (e.g., $\mathbb{Z}$ or rings of algebraic integers) is a non-primitive Dedekind prime ring, due to the absence of faithful simple modules.
• The matrix ring $M_n(R)$ over a Dedekind domain $R$ is also non-primitive Dedekind prime; every ideal in $M_n(R)$ is of form $M_n(I)$ for $I$ invertible in $R$.
• Maximal orders $\mathcal{O}$ in finite-dimensional division algebras $D$ over global fields $K$ form another class of non-primitive Dedekind prime rings.

3. Ideals, Lattice Structure, and Invariants

Let $A$ be a non-primitive Dedekind prime ring. Its ideal-theoretic and module-theoretic architecture is as follows:

• Every nonzero two-sided ideal $B$ is invertible: there exists $B^{-1} \subset Q$ such that $B B^{-1} = B^{-1} B = A$.
• Maximal invertible ideals $\mathcal{P}(A)$ completely classify two-sided ideals. For $B \subset A$ nonzero:

$$B = \prod_{P \in \mathcal{P}(A)} P^{e_P}, \quad e_P \geq 1$$

• The two-sided ideal lattice is free abelian on $\mathcal{P}(A)$.
• The ACC (ascending chain condition) holds for one-sided annihilators; the DCC applies to chains of invertible ideals.
• Uniform dimension (Goldie dimension) is determined by $Q \cong M_n(D)$, so $\operatorname{u.dim}(A) = n$.

4. Module Decomposition: Primary Components and Singularity

Module categories over $A$ mirror the decomposition properties familiar from commutative Dedekind domains but generalized to the noncommutative setting. For any right $A$-module $M$ and maximal invertible ideal $P \in \mathcal{P}(A)$,

$$M(P) = \{ m \in M \mid m P^k = 0 \text{ for some } k \}$$

Every module $M$ splits as a direct sum of its $P$–primary components:

$M \cong \bigoplus_{P \in \mathcal{P}(A)} M(P)$

There are no nonzero homomorphisms between distinct $P$–primary components, and torsion-theoretic principles from commutative theory carry over precisely.

5. Classification of Singular Generalized Bassian Modules

Singular modules are those whose elements are annihilated by nonzero ideals; generalized Bassian modules satisfy the condition that every injective homomorphism $M \to M/N$ implies $N$ is a direct summand.

Over non-primitive Dedekind prime rings, indecomposable injective singular modules are sharply classified:

• For $M$ with annihilator a single maximal ideal $P$, $M$ is uniserial: its submodule lattice is a chain.
• $M$ is non-cyclic, with no maximal proper submodule.
• All proper submodules are cyclic of finite length, comprising a countable ascending chain

$$0 = X_0 \subset X_1 \subset \cdots,\quad X_k / X_{k-1} \cong S \quad (\text{fixed simple}) \text{ for } k \geq 1$$

• Every proper homomorphic image of $M$ is again isomorphic to $M$.

These indecomposable injective singular modules act as noncommutative analogues of Prüfer $p$–groups, governing $P$–primary divisibility and torsion phenomena (Tuganbaev, 22 Jan 2026).

6. Context and Distinguishing Features

Non-primitive Dedekind prime rings form the class of one-dimensional, hereditary, noetherian, prime suborders inside simple artinian rings, with direct analogues to the theory of commutative Dedekind domains. The absence of faithful simple modules ensures these rings are non-artinian, catalyzing a rigid yet highly tractable ideal and module theory. The main invariants—uniform dimension, ideal factorization, and submodule lattice properties—enforce a structure in which decomposition, injectivity, and divisibility phenomena have clear and unique module-theoretic representatives. This framework provides fertile ground for a noncommutative analogue of singular module classification as explored in depth by Tuganbaev (Tuganbaev, 22 Jan 2026).