[Paper Review] Matlis' semi-regularity in trivial ring extensions issued from integral domains

Khalid Adarbeh, S. Kabbaj|arXiv (Cornell University)|Apr 11, 2016

Rings, Modules, and Algebras34 references3 citations

TL;DR

This paper investigates the transfer of Matlis' semi-regularity (also known as IF-ring property) and related coherence conditions in trivial ring extensions $A \ltimes E$ where $A$ is an integral domain and $E$ is an $A$-module. The key contribution is a characterization of semi-regularity in such extensions via the divisorial ideal condition (DAC) on $E$, yielding new families of semi-regular rings that are neither von Neumann regular nor quasi-Frobenius.

ABSTRACT

This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular ring (also known as IF-ring) along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the paper, we put the new results in use to enrich the literature with new families of examples subject to semi-regularity.

Motivation & Objective

• To investigate the transfer of Matlis' semi-regularity (IF-ring property) in trivial ring extensions $A \ltimes E$ where $A$ is an integral domain.
• To determine necessary and sufficient conditions for the trivial extension $A \ltimes E$ to be semi-regular, especially when $A$ is coherent or integrally closed.
• To enrich the literature with new families of semi-regular rings that are not Noetherian or reduced, thus extending beyond classical classes like von Neumann regular or quasi-Frobenius rings.
• To clarify the role of fp-injectivity and divisibility of $E$ as necessary conditions for semi-regularity in $A \ltimes E$.
• To provide constructive examples of semi-regular trivial extensions using Prüfer domains, divisorial domains, and non-standard uniserial modules.

Proposed method

• Utilizes the trivial ring extension construction $R = A \ltimes E$, where $A$ is an integral domain and $E$ is an $A$-module, to study homological properties.
• Applies the characterization of semi-regular rings as coherent rings that are self-fp-injective (i.e., $R$ is fp-injective as an $R$-module).
• Employs the divisorial ideal condition (DAC): for every finitely generated ideal $I$ of $A$, $(I : (I : J)) = J_v$ for all finitely generated ideals $J$, to characterize semi-regularity.
• Leverages known results on coherent domains, Prüfer domains, and valuation rings to construct examples where $A \ltimes \frac{Q(A)}{I}$ is semi-regular.
• Analyzes the structure of modules $E$ such as $\frac{Q(A)}{I}$ and non-standard uniserial modules over valuation domains to verify semi-regularity.
• Uses homological tools including $\operatorname{Ext}^1_R(M, E) = 0$ for finitely presented $M$ to verify fp-injectivity and thus semi-regularity.

Experimental results

Research questions

• RQ1 Under what conditions is the trivial ring extension $A \ltimes E$ semi-regular when $A$ is an integral domain?
• RQ2 What role does the divisorial ideal condition (DAC) play in determining semi-regularity of $A \ltimes E$?
• RQ3 Can semi-regular trivial extensions be constructed that are neither von Neumann regular nor Noetherian?
• RQ4 How does the fp-injectivity or divisibility of $E$ relate to the semi-regularity of $A \ltimes E$?
• RQ5 What types of modules $E$ (e.g., $Q(A)/I$, non-standard uniserial modules) yield semi-regular trivial extensions?

Key findings

• The trivial extension $A \ltimes \frac{Q(A)}{I}$ is semi-regular if and only if $\frac{Q(A)}{I}$ satisfies the divisorial ideal condition (DAC).
• If $A$ is a coherent domain that is not a field and $I$ is a nonzero finitely generated fractional ideal, then $A \ltimes \frac{Q(A)}{I}$ is semi-regular if $A$ is Prüfer or divisorial.
• For an integrally closed coherent domain $A$, $A \ltimes \frac{Q(A)}{A}$ is semi-regular if and only if $A$ is a Prüfer domain.
• There exist non-integrally closed, non-divisorial coherent domains $A$ such that $A \ltimes \frac{Q(A)}{I}$ is semi-regular, provided all finitely generated ideals of $A$ are divisorial.
• Non-standard uniserial modules $E$ over valuation domains can yield semi-regular trivial extensions $A \ltimes E$, even when $E$ is not isomorphic to a quotient $J/I$.
• The construction yields new examples of semi-regular rings that are neither von Neumann regular (due to zero divisors) nor quasi-Frobenius (due to non-Noetherian structure).