Trivial Ring Extensions: Definition and Applications

• Trivial ring extensions are defined by extending a ring R with an R-module M, yielding a new ring R ⨁ M where (0, M) forms a nilpotent ideal.
• They serve as accessible models to investigate module-theoretic properties, clarifying the structures of projective, injective, and flat modules.
• Applications include constructing non-reduced rings to test transfer principles in coherence, regularity, Gaussian properties, and Gorenstein conditions.

A trivial ring extension is a canonical construction in ring and module theory in which a (typically commutative) ring $R$ is extended by an $R$-module or $R$–$R$-bimodule $M$ to form a new ring $R \ltimes M$. This additive group is $R \oplus M$, with multiplication $(r,m)\cdot(r',m') = (rr',\; r m' + m r')$. Such extensions play a fundamental role in both commutative and non-commutative ring theory, providing a tractable class of non-reduced rings with controlled nilpotent structure, and feature prominently in the paper of homological, categorical, and homotopical properties of ring extensions and their associated module categories.

1. Formal Definition and Basic Properties

Let $R$ be a unital (possibly associative, not necessarily commutative) ring and $M$ an $R$-$R$-bimodule. The trivial ring extension (sometimes called the Nagata idealization or split null extension) is defined as: $$R \ltimes M = \{\, (r,m) \mid r\in R,\, m\in M \,\}$$ with addition and multiplication given by: $$(r,m) + (r',m') = (r+r',\, m + m'), \quad (r,m) \cdot (r',m') = (rr',\, r m' + m r')$$ The element $(0, m)$ is central only if $M$ is symmetric as a bimodule. The set $0 \oplus M$ forms a two-sided nilpotent ideal, and $(1, 0)$ is the multiplicative identity. The quotient $R\ltimes M / (0 \oplus M) \cong R$ recovers the base ring.

For $n$-trivial extensions, as in Benkhadra–Bennis–García Rozas (Benkhadra et al., 2019), given $R$ and an $n$-tuple $M = (M_1,\ldots,M_n)$ of $R$-bimodules, and maps $P_{i,j}: M_i\otimes_R M_j \to M_{i+j}$ (for $i+j\le n$) satisfying associativity conditions, the $n$-trivial extension ring $S$ is $R \oplus M_1 \oplus \cdots \oplus M_n$ with induced multiplication.

The classical trivial extension is the $n = 1$ case, $R \ltimes M$ (Bennis et al., 2016, Anderson et al., 2016).

2. Module-Theoretic and Homological Structure

A left $R \ltimes M$-module can be equivalently described as an $R$-module $X$ together with an $R$-linear homomorphism $\alpha: M\otimes_R X \to X$ with $\alpha \circ (M\otimes\alpha)=0$. This reflects the relation $(0,m)^2 = 0$. Projective, injective, and flat $R \ltimes M$-modules can be explicitly characterized:

• Projectives are of the form $T(P) = (P\oplus M\otimes_R P \to P)$ where $P$ is projective over $R$.
• Injectives as $H(E)=[\operatorname{Hom}_R(M,E)\to E]$ with $E$ injective over $R$.
• Flatness and Gorenstein properties require additional bimodule hypotheses (Mao, 2023).

Homological dimensions and categorical invariants such as the singularity category $D_{\mathrm{sg}}$ and Gorenstein defect category $D_{\mathrm{def}}$ are intensely studied in the context of trivial extensions, as these categories often reduce to their counterparts for the base ring $R$ under appropriate Tor-vanishing and nilpotence conditions on $M$ (Qin, 19 Mar 2024).

3. Ring-Theoretic Properties and Transfer Principles

Trivial ring extensions transfer and reflect a wide spectrum of ring-theoretic properties, depending on the structure of $R$ and $M$:

• Prime and maximal ideals: Every prime ideal of $R \ltimes M$ has the form $P \oplus M$ for $P$ prime in $R$; similar for maximal ideals.
• Coherence and regularity: $R \ltimes M$ is coherent if $R$ is coherent, $M$ is torsion coherent and for all $x \in R$, $\operatorname{Ann}_M(x)$ is finitely generated (Adarbeh et al., 2016).
• Weak dimension and global dimension: If $R$ is an fqf-ring, then $\mathrm{w.gl.d}(R \ltimes M)$ is in $\{0,1,\infty\}$, and the finitistic weak dimension is $0,1$ or $2$ (Couchot, 2015).
• Prüfer, Bézout, and Gaussian properties: Sharp transfer theorems relate the status of $R$ and $M$ to the corresponding property for $R \ltimes M$, e.g. $R \ltimes M$ is Gaussian iff $R$ is Gaussian and $aM = a^2 M$ for all $a \in R$ (Couchot, 2015, Bakkari et al., 2008).

These transfer phenomena provide a calculable testbed to produce new examples of rings with prescribed homological or ideal-theoretic behaviors, including non-coherent, non-Noetherian, and non-reduced rings.

4. Extensions, Generalizations, and Categorical Aspects

The construction extends naturally to $n$-trivial extensions and to more involved settings:

• $n$-trivial extensions: For $M = (M_1, \ldots, M_n)$ with specific multiplications $P_{i,j}$, the $n$-trivial extension $R \oplus_n M$ encapsulates higher-degree analogs and supports graded structures—$\mathbb{N}_0$-graded, $\mathbb{Z}/(n+1)$-graded, or graded by truncated monoids (Anderson et al., 2016, Benkhadra et al., 2019).
• Triangular matrix algebras: Triangular matrix rings $\mathrm{Tri}(A,B;M)$ can be realized as trivial extensions of $A\oplus B$ by an $(A\oplus B)$-bimodule $M$ with explicit action (Bennis et al., 2016).
• Quivers and relations: The trivial extension of a finite-dimensional algebra $A$ by its standard dual $D(A)$ admits an explicit quiver and relations description. The Gabriel quiver of $T(A)=A\ltimes D(A)$ extends the quiver of $A$ by dual arrows for socle elements, and the relations are encoded in a combinatorial fashion (Fernandez et al., 2022).

Categorical perspectives include the module category of $R\ltimes M$, equivalences of derived and singularity categories, and characterizations in terms of functor categories (right/left $n$-trivial extensions of categories by endofunctor families) (Benkhadra et al., 2019).

5. Homological and Gorenstein Aspects

Gorenstein projective, injective, and flat module categories over $R\ltimes M$ have been fully described in terms of generalized compatible and cocompatible bimodule structures (Mao, 2023). Explicit criteria relate the existence of complete resolutions and the vanishing of certain derived functors (Tor, Ext) to the corresponding properties for $M$ over $R$:

• $(X,\alpha)$ is Gorenstein projective over $R\ltimes M$ iff the sequence $M\otimes_R M\otimes_R X \to M\otimes_R X \to X$ is exact and $\mathrm{coker}(\alpha)$ is Gorenstein projective over $R$.
• Gorenstein injective and flat modules are similarly characterized using appropriate exactness conditions on associated complexes.

Trivial extensions are instrumental in studying the ascent and descent of Gorensteinness and related invariants in extension settings (Mao, 2023, Qin, 19 Mar 2024).

6. Cohen–Macaulayness, CS-Rings, and Semi-Regularity

The trivial extension $R\ltimes M$ serves as a crucial testing ground for the behavior of non-Noetherian analogs of regularity:

• Cohen–Macaulayness: $R\ltimes M$ is Cohen–Macaulay (in the Hamilton–Marley sense) if and only if $R$ is Cohen–Macaulay and every $R$-regular sequence is weakly $M$-regular (Mahdikhani et al., 2017).
• CS-rings: $R\ltimes M$ is a CS ring if and only if $\operatorname{Ann}_R(M)$ is a direct summand and CS as a ring, and $M$ is weakly IN (annihilator sum) (Kourki et al., 2021).
• Semi-regularity (IF-ring property): For $A$ a domain, $A\ltimes E$ is semi-regular iff $A$ is a field and $E\cong A$, or $A$ is coherent, $E$ divisible, torsion, coherent, satisfying double annihilator condition, and appropriate annihilators are finitely generated (Adarbeh et al., 2016).

This provides systematic control over the appearance of various forms of regularity, with direct application to the construction of rings with prescribed regularity failures.

7. Applications and Open Problems

Trivial ring extensions have deep applications in several areas:

• Prüfer, arithmetical, and Gaussian ring construction: Generating examples and counterexamples bearing on the Bazzoni–Glaz conjecture (weak dimension in Gaussian rings) and the Kaplansky–Tsang–Glaz–Vasconcelos content ideal conjecture via idealizations $R\ltimes M$ (Bakkari et al., 2008).
• Singularity theory and categorical equivalence: Reduction of singularity and Gorenstein defect categories under split extensions by nilpotent bimodules with suitable Tor vanishing, aiding classification of singularities in finite-dimensional algebras (Qin, 19 Mar 2024).
• Extension of factorization and divisibility theory: Transfer and refinement of ACCP, atomicity, and bounded-factorization phenomena in the $n$-trivial extension setting, with open questions on U-factorizations and higher-degree indecomposability (Anderson et al., 2016, Benkhadra et al., 2019).

Important open problems persist regarding the precise characterization of U-factorization, the behavior under more general (e.g., nontrivial) extensions, and the interplay with other classical properties (valuation, ZPI, etc.), particularly in higher $n$-trivial constructs.

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References:

• (Bakkari et al., 2008) Bakkari, Kabbaj, Mahdou, "Trivial extensions defined by Prufer conditions"
• (Couchot, 2015) Couchot, "Gaussian trivial ring extensions and fqp-rings"
• (Anderson et al., 2016) Benkhadra, Bennis, García Rozas, "On n-Trivial Extensions of Rings"
• (Bennis et al., 2016) Birkenmeier, Ortega, Wang, "Derivations and the first cohomology group of trivial extension algebras"
• (Adarbeh et al., 2016) Adarbeh, Kabbaj, "Matlis' semi-regularity in trivial ring extensions issued from integral domains"
• (Mahdikhani et al., 2017) Mahdikhani, Sahandi, Shirmohammadi, "Cohen-Macaulayness of trivial extensions"
• (Benkhadra et al., 2019) Benkhadra, Bennis, García Rozas, "The category of modules on an n-trivial extension: the basic properties"
• (Kourki et al., 2021) Ünver, Savaş, "On Two Classes of Modules Related to CS Trivial Extensions"
• (Fernandez et al., 2022) Białkowski, Skowroński, "Characterisations of trivial extensions"
• (Mao, 2023) Mao, "Gorenstein projective, injective and flat modules over trivial ring extensions"
• (Qin, 19 Mar 2024) Lin, Zhang, Zhou, "Singular equivalences induced by ring extensions"