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 $A$0, with every nonzero (two-sided) ideal of $A$1 invertible in $A$2. $A$3 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$4–$A$5-bimodules.

Key features include:

• Every essential one-sided ideal contains a nonzero two-sided ideal.
• The set $A$6 of maximal invertible ideals serves as “prime ideals,” with every nonzero ideal factoring uniquely as a product of elements of $A$7.
• $A$8 is a right and left Goldie ring; its uniform dimension equals the matrix size $A$9 in $M\to M/N$0.
• All nonzero (one-sided) ideals are projective and exhibit constant rank one over $M\to M/N$1.
• 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., $M\to M/N$2 or rings of algebraic integers) is a non-primitive Dedekind prime ring, due to the absence of faithful simple modules.
• The matrix ring $M\to M/N$3 over a Dedekind domain $M\to M/N$4 is also non-primitive Dedekind prime; every ideal in $M\to M/N$5 is of form $M\to M/N$6 for $M\to M/N$7 invertible in $M\to M/N$8.
• Maximal orders $M\to M/N$9 in finite-dimensional division algebras $N$0 over global fields $N$1 form another class of non-primitive Dedekind prime rings.

3. Ideals, Lattice Structure, and Invariants

Let $N$2 be a non-primitive Dedekind prime ring. Its ideal-theoretic and module-theoretic architecture is as follows:

• Every nonzero two-sided ideal $N$3 is invertible: there exists $N$4 such that $N$5.
• Maximal invertible ideals $N$6 completely classify two-sided ideals. For $N$7 nonzero:

$N$8

• The two-sided ideal lattice is free abelian on $N$9.
• 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 $M$0, so $M$1.

4. Module Decomposition: Primary Components and Singularity

Module categories over $M$2 mirror the decomposition properties familiar from commutative Dedekind domains but generalized to the noncommutative setting. For any right $M$3-module $M$4 and maximal invertible ideal $M$5,

$M$6

Every module $M$7 splits as a direct sum of its $M$8–primary components:

$M$9

There are no nonzero homomorphisms between distinct $N$0–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 $N$1 implies $N$2 is a direct summand.

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

• For $N$3 with annihilator a single maximal ideal $N$4, $N$5 is uniserial: its submodule lattice is a chain.
• $N$6 is non-cyclic, with no maximal proper submodule.
• All proper submodules are cyclic of finite length, comprising a countable ascending chain

$N$7

• Every proper homomorphic image of $N$8 is again isomorphic to $N$9.

These indecomposable injective singular modules act as noncommutative analogues of Prüfer $M$0–groups, governing $M$1–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).