Trivial Ring Extensions

• Trivial ring extensions are constructions that combine a ring with an auxiliary module to form a new ring featuring a square-zero ideal.
• They serve as a framework to analyze structural and homological properties, including Noetherian conditions, Prüfer and Gaussian behaviors, and singularity categories.
• Applications extend to graded structures and n-trivial extensions, providing insight into conjectures and module-theoretic phenomena in modern algebra.

A trivial ring extension, also known as an idealization or Nagata idealization, is a fundamental construction in commutative and noncommutative algebra, synthesizing the structure of a ring with an auxiliary module into a new ring in which the module appears as a square-zero ideal. This construction serves as a unifying template for producing diverse classes of rings with prescribed homological, structural, or categorical properties, and is pervasive in the paper of Prüfer-type rings, module-theoretic conditions (CS, fqp, IF, CM), singularity categories, Gorenstein homology, and extensions to higher layers via $n$-trivial extensions.

1. Definition and Foundational Properties

Let $R$ be a (commutative or associative) ring with $1$ and $M$ an $R$-module (or, in the noncommutative case, an $R$-$R$-bimodule). The trivial ring extension $R\ltimes M$ is the abelian group $R \oplus M$ endowed with multiplication

$(r, m) \cdot (s, n) = (r s, r n + s m)$

for all $r, s \in R$, $m, n \in M$ (Bakkari et al., 2008, Fernandez et al., 2022, Mao, 2023, Couchot, 2015). The set $0 \ltimes M$ is an ideal of square zero; $R$ embeds as $R \times \{0\}$ and the quotient $R\ltimes M / (0 \ltimes M) \cong R$. The prime spectrum of $R \ltimes M$ reflects that of $R$ and $\operatorname{Supp} M$, with $$\operatorname{Spec}(R\ltimes M) \cong \operatorname{Spec} R \; \sqcup \; \{q \ | \ q \text{ is of type } P\ltimes M\}$$ (Mahdikhani et al., 2017). Every ideal has the form $I \ltimes N$ with $I \subseteq R$ an ideal, $N \subseteq M$ an $R$-submodule, and $IM \subseteq N$.

Extensive generalization is achieved in the $n$-trivial extension $$T_n(R, (M_i)_{1 \le i \le n}) = R \oplus M_1 \oplus \cdots \oplus M_n$$ with multiplication governed by a family of bilinear maps $\varphi_{i, j}: M_i \times M_j \to M_{i+j}$ satisfying well-posed associativity and commutativity axioms. For $n=1$, this reduces to the classical $R\ltimes M$ construction (Anderson et al., 2016, Benkhadra et al., 2019).

2. Structural and Homological Properties

The trivial extension is non-reduced unless $M=0$. If $R$ is commutative, $0 \ltimes M$ will consist entirely of nilpotent elements. For $R$ Noetherian and $M$ finitely generated, the dimension formula is

$\dim (R\ltimes M) = \max\{\dim R, \dim M\}$

with $\dim M = \dim (R/\operatorname{Ann} M)$ (Mahdikhani et al., 2017). The radical layers satisfy $\Nil(R\ltimes M ) = \Nil(R)\ltimes M$, $\Jac(R\ltimes M) = \Jac(R)\ltimes M$ (Anderson et al., 2016).

Finiteness and hereditary properties transfer partially. $R\ltimes M$ is Noetherian (resp. Artinian) if and only if $R$ is Noetherian (resp. Artinian) and $M$ is finitely generated (resp. finite length). In general, chains of ideals and modules induce complex behaviors in the extension, reflecting but vastly extending the fine structure of $R$ and $M$ (Couchot, 2015). For local rings, the trivial extension is again local.

3. Interaction with Prüfer-, Gaussian-, and Arithmetical-Type Conditions

Prüfer and related conditions finely stratify on trivial ring extensions. In the classical domain setting, the following hierarchy holds: semihereditary $\Rightarrow$ weak dimension $\le 1$ $\Rightarrow$ arithmetical $\Rightarrow$ Gaussian $\Rightarrow$ Prüfer; these implications are strict outside domains (Bakkari et al., 2008, Couchot, 2015). Sharp transfer theorems are established:

Prüfer and Gaussian Properties:

• $R = A \ltimes B$ with $A \subseteq B$ domains, $K$ the quotient field of $A$:
  • $R$ is Gaussian $\Longleftrightarrow$ $R$ is Prüfer $\Longleftrightarrow$ $A$ is Prüfer domain and $K \subseteq B$.
  • $R$ is arithmetical $\Longleftrightarrow$ $A$ is Prüfer and $K = B$.
  • The weak dimension satisfies $\operatorname{w.dim}(R) = \infty$ unless $A$ is a field or $M=0$.

Gaussian Criterion (arbitrary base):

$R = A \propto E$ is Gaussian if and only if $A$ is Gaussian and $aE = a^2E$ for all $a \in A$ (Couchot, 2015).

Under suitable topological splitting ($\operatorname{pSpec}(A)$ totally disconnected), $R$ is Bézout (all finitely generated ideals principal) if and only if $A$ is Bézout, each localization $A_P$ with $P \in \operatorname{Supp} E$ is a domain, and $E$ is locally FP-injective with all finitely generated submodules cyclic.

(fqp)- and (fqf)-Rings:

$fqp$-rings (every finitely generated ideal quasi-projective) and $fqf$-rings (every finitely generated ideal flat over the quotient by its annihilator) are characterized locally; in trivial extensions, they exhibit the following dichotomy:

• $R = A \propto E$ is local fqp iff $A$ is fqp and either $A$ is a valuation domain and $E$ is divisible uniserial, or $N^2=0$, $E$ and $N$ divisible/torsion-free over $A/N$ (Couchot, 2015).

4. Cohen–Macaulayness, CS-Modules, and Semi-Regularity

Cohen–Macaulayness in the sense of Hamilton–Marley (via weakly proregular parameter sequences) is transferred via: $R\ltimes M \text{ is CM} \iff R \text{ is CM and every %%%%84%%%%-regular sequence is a weak %%%%85%%%%-regular sequence}$ In the Noetherian local case, $R\ltimes M$ is CM if and only if $R$ is CM and $M$ is maximal CM (Mahdikhani et al., 2017).

CS (Extending) Rings:

$R\propto M$ is CS if and only if $\operatorname{Ann}_R(M)$ is a direct summand of $R$ and a CS ring, and $M$ is weakly IN or strongly CS—a module-theoretic strengthening of the extending/annihilator property (Kourki et al., 2021).

Semi-Regularity (IF-Rings):

For $A$ a domain, $R = A\ltimes E$ is semi-regular (Matlis IF-ring) iff $A$ is coherent, $E$ is divisible torsion coherent (fp-injective), $\operatorname{Ann}_E(x)$ finitely generated for every $x\in A$ and a double annihilator condition holds in both $A$ and $E$ (Adarbeh et al., 2016).

5. Homological and Categorical Behavior

Gorenstein Projective/Injective/Flat Modules:

Classification over $R\ltimes M$ is realized using generalized compatible and cocompatible bimodule conditions on $M$ (Mao, 2023):

• $(X, \alpha)$ is Gorenstein projective as an $R\ltimes M$-module iff:
  1. The complex $$M \otimes_R M \otimes_R X \xrightarrow{M\otimes \alpha} M \otimes_R X \xrightarrow{\alpha} X$$ is exact.
  2. $\operatorname{coker}\alpha$ is Gorenstein projective over $R$. Analogous characterizations hold for Gorenstein injective and flat modules, leveraging the corresponding Hom-complexes.

Singularity Categories and Gorenstein Defect:

For finite-dimensional $k$-algebras $R$ and a bimodule $M$ with finite projective dimension over $R^e$, $M^{\otimes_R p}=0$ for some $p$, and vanishing higher $\operatorname{Tor}^i_R(M, M^{\otimes_R j})$, there exist equivalences

$$\mathcal{D}_{\operatorname{sg}}(R) \simeq \mathcal{D}_{\operatorname{sg}}(R\ltimes M),\quad \mathcal{D}_{\operatorname{def}}(R)\simeq \mathcal{D}_{\operatorname{def}}(R\ltimes M)$$

providing computational reduction for singularities and Gorenstein-locus behavior (Qin, 19 Mar 2024).

6. Quivers with Relations and Trivial Extensions

For a finite-dimensional algebra $A=kQ/I$ and bimodule $M$ with socle basis, the trivial extension $A\ltimes M$ allows completely explicit quiver-with-relations presentations:

• The quiver $Q_{T}$ retains the original vertices, all original arrows, and adds new arrows $B_{p_i}$ corresponding to socle basis paths $p_i$ (from their targets to sources).
• Relations are given by: original relations; vanishing of non-elementary paths; and combinatorial cycle-weight conditions ensuring all elementary cycles based at a vertex are proportional. An explicit criterion (Wakamatsu's theorem) characterizes when two algebras have isomorphic trivial extensions based on admissible cuts in the quiver $Q_{T}$ (Fernandez et al., 2022).

7. $n$-Trivial Extensions and Graded Structures

$n$-trivial extensions, $T_n(R, (M_i)_{1 \le i \le n})$, generalize the construction to multiple layers of modules with higher-degree multiplications given by bilinear maps $\varphi_{i, j}$ (Anderson et al., 2016, Benkhadra et al., 2019). This setting supports rich graded structures ($\mathbb{N}$-grading, cyclic grading, truncated monoid grading), and the transfer of Noetherian, Artinian, reduced, and local/valuation-type properties depends recursively on the base ring and the layers $M_i$. Factorization and divisibility conditions require compatibility among the $M_i$, with phenomena such as atomicity, ACCP, and U-Factorization closely tracking module-theoretic properties.

Categorically, the abelian category of modules over an $n$-trivial extension is constructed via extensions of additive covariant endofunctors, providing explicit classification of projective, injective, and flat modules. The self-injective and global dimensions of $T_n(R, (M_i))$ can often be directly related to those of $R$ and the maximal module $M_n$ (Benkhadra et al., 2019).

8. Illustrative Examples and Applications

• $R = D \ltimes K$, $D$ a Prüfer domain, $K$ its field of fractions, gives a non-Noetherian, non-coherent arithmetical ring of infinite weak dimension (Bakkari et al., 2008).
• $R = K \ltimes L$, $K \subsetneq L$ a field extension, yields Gaussian but non-arithmetical trivial extensions.
• $R = \mathbb{Z}_{(2)} \ltimes \mathbb{R}$: Gaussian, non-arithmetical, non-coherent, infinite weak dimension.
• $R \ltimes M$, $M$ the total ring of quotients of $R$, $R$ CS iff $R \ltimes M$ CS (Kourki et al., 2021).
• Morita context rings $$\Lambda = \begin{pmatrix}A & V \ U & B\end{pmatrix}$$ with zero maps are isomorphic to $(A \times B) \ltimes (U \oplus V)$.
• $$T_n(R, R^{\oplus r}) \cong R[x_1, \dotsc, x_r]/(\text{all monomials of degree } n+1)$$, relating trivial extensions to truncated polynomial algebras (Anderson et al., 2016).

9. Connections to Conjectures and Open Problems

Trivial ring extensions serve as canonical sources of counterexamples and verification grounds for several open conjectures:

• Bazzoni–Glaz Weak Dimension Conjecture: All non-Noetherian Gaussian trivial extensions constructed have weak dimension $0$, $1$, or $\infty$ (Bakkari et al., 2008).
• Kaplansky–Tsang Content Conjecture: Numerous non-Noetherian, non-arithmetical trivial extensions are pseudo-arithmetical (all Gaussian polynomials have locally principal content ideal). An open conjecture posits that pseudo-arithmeticality is equivalent to the local irreducibility of the zero ideal (Bakkari et al., 2008).

Extensions to higher $n$-trivial settings invoke new phenomena regarding the homogeneity of ideals, $\varphi$-indecomposability, and the classification of atomicity and factorization properties, opening additional lines of inquiry in the interaction of algebraic and categorical invariants (Anderson et al., 2016).

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Trivial ring extensions and their higher analogs, via their rich interactions with module and homological theory, provide laboratories to both realize and distinguish a wide array of algebraic, categorical, and homological phenomena, uniting classical and recent advances in commutative algebra, representation theory, and homological algebra.