Let f: A → B be a ring homomorphism, and let J be an ideal of B. In this article, we study the amalgamation of A with B along J with respect to f (denoted by A ⋈fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the A + XB[X], the A + XB[[X]] and the D + M constructions). In particular, we completely describe the prime spectrum of the amalgamation A ⋈fJ and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen–Macaulay ring or a Gorenstein ring.

New Algebraic Properties of an Amalgamated Algebra Along an Ideal

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ 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 $B\times C$ given by $$A\bowtie^{f,g}(J,J'):=\big\{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big\}.$$ 
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.

Bi-Amalgamated algebras along ideals

Let f : A ! B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of the property of coherence to the amalgamation A ▷◁ f J. We provide necessary and sufficient conditions for A ▷◁ f J to be a coherent ring.

http://bims.iranjournals.ir/article_637_ee6424db1fd55f61b941f1ae5f86a13b.pdf

Coherence in amalgamated algebra along an ideal

Let $f: A\rightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we investigate the transfert of the property of coherence to the amalgamation $A\bowtie^{f}J$. We provide necessary and sufficient conditions for $A\bowtie^{f}J$ to be a coherent ring.

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.

Bi-amalgamations subject to the arithmetical property

This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prufer conditions.

M. Jarrar

Papers

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

Prufer conditions in an amalgamated duplication of a ring along an ideal