Non-Primitive Dedekind Prime Rings

• Non-primitive Dedekind prime rings are noetherian, prime rings whose every nonzero ideal is invertible in a simple Artinian fraction ring while lacking faithful simple modules.
• Their structure is defined by bounded hereditary properties and maximal orders in central simple algebras, as emphasized by the Lenagan–Robson theorem.
• Invertible ideal theory in these rings guarantees unique factorization of nonzero ideals, which is key to understanding their module and singular injective structures.

A non-primitive Dedekind prime ring is an associative unital ring $A$ that is noetherian and prime, where every nonzero two-sided ideal of $A$ is invertible within its simple Artinian ring of fractions $Q = A[S^{-1}] \cong M_n(D)$ for some $n \ge 1$, $D$ a division ring, yet $A$ itself lacks faithful simple right (or left) modules and is not a simple Artinian ring. These rings form a distinguished subclass of hereditary noetherian prime (HNP) rings characterized by their ideal-theoretic properties, module structure, and order-theoretic interpretation in central simple algebras (Tuganbaev, 22 Jan 2026).

1. Precise Definitions

Let $A$ be an associative unital ring. The following definitions establish the conceptual framework:

• Prime Ring: $A$ is prime if for any nonzero two-sided ideals $B, C \subseteq A$, the product $BC \ne 0$.
• Noetherian: $A$ satisfies the ascending chain condition on both right and left ideals.
• Invertible Ideal: An ideal $I \subseteq A$ is invertible (in $Q$) if there exists an $A$–$A$ subbimodule $I^{-1} \subseteq Q$ such that $I I^{-1} = I^{-1} I = A$ inside $Q$.

Dedekind Prime Ring: $$A\text{ is Dedekind prime} \quad \Longleftrightarrow \quad \begin{cases} A \text{ is noetherian and prime}, \ Q = \operatorname{Frac}(A) \cong M_n(D), \ \text{all nonzero ideals of } A \text{ are invertible in } Q \end{cases}$$

Non-primitive: $$A \text{ is non-primitive} \quad \Longleftrightarrow \quad A \text{ has no faithful simple right (or left) modules.}$$

Formally, a non-primitive Dedekind prime ring $A$ is a noetherian prime ring with simple Artinian ring of fractions $Q$ such that every nonzero two-sided ideal of $A$ is invertible in $Q$ and $A$ itself is not a simple Artinian ring (Tuganbaev, 22 Jan 2026).

2. Structural Results and Classification

Non-primitive Dedekind prime rings are bounded HNP rings that are maximal orders in their simple Artinian fraction rings. Two key theorems form the basis of their structural understanding:

• Lenagan–Robson Theorem: Any hereditary noetherian prime (HNP) ring $A$ is either primitive or bounded. If $A$ is both primitive and bounded, then it is simple Artinian. Non-primitive HNP rings are precisely bounded, non-Artinian HNP rings.
• Characterization Theorem: For $A$ a noetherian prime ring with semisimple Artinian ring of fractions $Q$, the following are equivalent:
  1. Every nonzero ideal of $A$ is an invertible $Q$-ideal.
  2. $A$ is hereditary and every projective right (or left) ideal is two-sided.
  3. $A$ is a (maximal) order in the central simple algebra $Q$.

In summary, non-primitive Dedekind prime rings are bounded hereditary noetherian prime rings that are maximal orders in a simple Artinian algebra, but lack faithful simple modules, distinguishing them from the simple Artinian case.

3. Representative Examples

A range of commutative and noncommutative constructions yield non-primitive Dedekind prime rings:

• Commutative Dedekind Domains: Any Dedekind domain $R$ which is not a field (e.g., $\mathbb{Z}$, rings of integers in number fields).

• Matrix Rings: $A = M_n(R)$, with $n \ge 1$, $R$ a Dedekind domain. For $n > 1$, primitivity fails unless $\operatorname{Frac}(R)$ is a field acting faithfully on a single row.
• Maximal Orders in Division Algebras: Given a global or local field $K$ and central division algebra $D$ over $K$, a maximal $O_K$-order $A \subseteq D$ is typically a non-primitive Dedekind prime ring (unless $D = K$ and $A = O_K$).
• Hereditary Orders in Separable Algebras: Any hereditary order in a separable algebra over a Dedekind domain that is not simple Artinian.

Example Type	Ring Structure	Primitivity
Dedekind domain ($R$)	Commutative	Non-primitive (if not a field)
Matrix ring ($M_n(R)$, $n>1$)	Noncommutative	Non-primitive
Maximal order in division algebra	Central simple	Non-primitive (typically)
Hereditary order in separable alg.	Noncommutative	Non-primitive

4. Key Properties and Invariants

Central properties and invariants of non-primitive Dedekind prime rings include:

• Noetherian, Hereditary, Prime, Semiprime: Inferred from definitions and structural theorems; every Dedekind prime ring is a hereditary noetherian prime ring.
• Goldie Dimension: $A$ has right Goldie dimension $u.\dim(A) = n$ when $Q \cong M_n(D)$.
• Boundedness: Every essential right ideal contains a nonzero two-sided ideal; singular right ideals are precisely those containing no regular elements.
• Invertible Ideal Theory: Each nonzero two-sided ideal $I \subseteq A$ is invertible; maximal invertible ideals $\mathcal{P}(A)$ play the role of prime divisors, allowing for unique factorization of ideals.
• Fraction Ring: $Q$ is a semisimple Artinian ring; every nonzero ideal of $A$ is essential as both a right and left ideal, therefore containing regular elements.
• Chain Conditions: $A$ satisfies the ascending chain condition (ACC) on right and left annihilator ideals.
• Module Theory: Singular (torsion) modules split into their primary components indexed by $P \in \mathcal{P}(A)$. Indecomposable injective singular modules are uniserial $P$-primary injectives as in Proposition 3.4 of (Tuganbaev, 22 Jan 2026); they possess a unique countable chain of cyclic submodules, with every nonzero quotient isomorphic to the module itself.

5. Ideal Theory: Factorization and Maximal Ideals

The theory of invertible ideals plays a foundational role:

• Definition (invertible ideal): $$I \subseteq A \text{ is invertible if } \exists\, I^{-1} \subseteq Q \text{ with } I I^{-1} = I^{-1} I = A$$
• Maximal Invertible Ideals $\mathcal{P}(A)$: These serve as analogues of prime divisors, particularly in the context of ideal factorization.
• Unique Factorization: Every nonzero ideal factors uniquely (up to order) into products of maximal invertible ideals in $\mathcal{P}(A)$. When $A$ is finitely generated over its center, these maximal ideals correspond to height-one prime ideals in the center.

6. Singular Injective Module Structure

Indecomposable injective singular modules in non-primitive Dedekind prime rings $A$ (with $P$-primary component) exhibit the following structure:

• Uniseriality: Such modules are non-cyclic and uniserial, possessing a complete chain

$0 = X_0 \subset X_1 \subset X_2 \subset \cdots$

where each quotient $X_k / X_{k-1}$ is simple.

• Self-isomorphism of Quotients: Every nonzero quotient of $E$ is isomorphic to $E$.
• Primary Decomposition: Every singular module splits into its primary components indexed by maximal invertible ideals.

References for Further Reading

• C. Faith, Algebra II, Springer–Verlag, 1976.
• K. R. Goodearl, R. B. Warfield, An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989.
• T. H. Lenagan, “Bounded hereditary noetherian prime rings,” Journal of the London Mathematical Society 6 (1973), 241–246.
• J. C. Robson, “Idealisers and hereditary noetherian prime rings,” Journal of Algebra 22 (1972), 45–81.
• A. Tuganbaev, “Generalized Bassian Modules over Non-primitive Dedekind Prime Rings” (Tuganbaev, 22 Jan 2026).