[Paper Review] Bi-Amalgamated algebras along ideals

S. Kabbaj, K. Louartit+1 more|arXiv (Cornell University)|Jul 26, 2014

Rings, Modules, and Algebras10 references3 citations

TL;DR

This paper introduces and studies bi-amalgamated algebras, a new construction of commutative rings formed as subrings of products $ B \times C $ via homomorphisms $ f: A \to B $, $ g: A \to C $, and ideals $ J \subseteq B $, $ J' \subseteq C $ with $ f^{-1}(J) = g^{-1}(J') $. The key contribution is a characterization of when such algebras are local, showing that $ A \bowtie^{f,g}(J,J') $ is local if and only if $ A $ is local and $ J \times J' \subseteq \text{Jac}(B \times C) $, generalizing earlier results on amalgamated duplications and algebras.

ABSTRACT

Let $f: A ightarrow B$ and $g: A ightarrow C$ be two commutative ring homomorphisms and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \emph{bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B imes C$ given by $$A\bowtie^{f,g}(J,J'):=\big\{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J imes J'\big\}.$$ This paper investigates ring-theoretic properties of \emph{bi-amalgamations} and capitalizes on previous works carried on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna-Finocchiaro-Fontana's amalgamations and duplications.

Motivation & Objective

• To generalize the construction of amalgamated algebras to a bimodule-like product structure using two homomorphisms and two ideals.
• To investigate the transfer of basic ring-theoretic properties (e.g., locality, prime ideals) to bi-amalgamated algebras.
• To establish connections between bi-amalgamations and pullbacks, particularly conductor squares.
• To extend known results on amalgamated duplications and algebras to a broader class of constructions.

Proposed method

• Define the bi-amalgamated algebra $ A \bowtie^{f,g}(J,J') $ as the subring of $ B \times C $ consisting of elements $ (f(a)+j, g(a)+j') $ for $ a \in A $, $ j \in J $, $ j' \in J' $, with $ f^{-1}(J) = g^{-1}(J') $.
• Use pullback constructions to show that bi-amalgamations arise naturally as fiber products under suitable conditions.
• Characterize the Jacobson radical of $ B \times C $ to analyze when $ J \times J' \subseteq \text{Jac}(B \times C) $, which is essential for locality.
• Establish a localization formula: $ (A \bowtie^{f,g}(J,J'))_P \cong A_p \bowtie^{f_p,g_p}(J_S, J'_{S'}) $ for prime ideals $ P $ containing $ J \times J' $, where $ S = f(A \setminus p) + J $, $ S' = g(A \setminus p) + J' $.
• Prove that $ A \bowtie^{f,g}(J,J') $ is local if and only if $ A $ is local and $ J \times J' \subseteq \text{Jac}(B \times C) $, generalizing known results on amalgamated duplications.
• Use the conductor square framework to relate the structure of $ A \bowtie^{f,g}(J,J') $ to its localizations and prime spectrum.

Experimental results

Research questions

• RQ1 When is the bi-amalgamated algebra $ A \bowtie^{f,g}(J,J') $ a local ring?
• RQ2 How do ring-theoretic properties such as locality, reducedness, and Noetherianity transfer from the base ring $ A $ and ideals $ J, J' $ to the bi-amalgamated algebra?
• RQ3 What is the prime ideal structure of $ A \bowtie^{f,g}(J,J') $, and how does it relate to the prime ideals of $ A $, $ B $, and $ C $?
• RQ4 Can bi-amalgamations be expressed as pullbacks, and if so, under what conditions?
• RQ5 How do localizations of $ A \bowtie^{f,g}(J,J') $ at prime ideals containing $ J \times J' $ behave, and what is their structure?

Key findings

• The bi-amalgamated algebra $ A \bowtie^{f,g}(J,J') $ is local if and only if the base ring $ A $ is local and the product ideal $ J \times J' $ is contained in the Jacobson radical of $ B \times C $.
• The construction generalizes both amalgamated algebras $ A \bowtie^f J $ and amalgamated duplications $ A \bowtie I $, recovering known results as special cases.
• Prime ideals of $ A \bowtie^{f,g}(J,J') $ containing $ J \times J' $ correspond bijectively to prime ideals $ p \subseteq A $ containing $ I_o = f^{-1}(J) = g^{-1}(J') $, with $ P = p \bowtie^{f,g}(J,J') $.
• The localization of $ A \bowtie^{f,g}(J,J') $ at such a prime $ P $ is isomorphic to $ A_p \bowtie^{f_p,g_p}(J_S, J'_{S'}) $, where $ S = f(A \setminus p) + J $, $ S' = g(A \setminus p) + J' $.
• The construction arises naturally as a pullback, and the fiber product structure allows for a conductor square description, linking the algebra to classical commutative algebra frameworks.
• The results are consistent with and extend recent literature on D’Anna-Finocchiaro-Fontana’s amalgamations, particularly in the context of prime spectra and ideal-theoretic properties.