--- title: Non-Primitive Dedekind Prime Rings url: https://www.emergentmind.com/topics/non-primitive-dedekind-prime-rings type: topic --- # Non-Primitive Dedekind Prime Rings 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 [2601.16201]. ## 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 [2601.16201]. ## 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 [2601.16201]; 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” [2601.16201]. Source: https://www.emergentmind.com/topics/non-primitive-dedekind-prime-rings