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On bi-amalgamated constructions

Federico Campanini, Carmelo Antonio Finocchiaro

Abstract

Let $f:A\longrightarrow B, g:A\longrightarrow C$ be ring homomorphisms and let $\mathfrak{b}$ (resp., $\mathfrak{c}$) be an ideal of $B$ (resp., $C$) satisfying $f^{-1}(\mathfrak{b})=g^{-1}(\mathfrak{c})$. Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring $$A\bowtie^{f,g}(\mathfrak{b},\mathfrak{c}) :=\{(f(a)+b, g(a)+c)\mid a\in A, b\in\mathfrak{b}, c\in \mathfrak{c} \}$$ of $B\times C$, called the bi-amalgamation of $A$ with $(B,C)$ along $(\mathfrak{b}, \mathfrak{c})$, with respect to $(f,g)$. This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D'Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante, by providing a deeper insigt on the ideal-theoretic structure of bi-amalgamations.

On bi-amalgamated constructions

Abstract

Let be ring homomorphisms and let (resp., ) be an ideal of (resp., ) satisfying . Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring of , called the bi-amalgamation of with along , with respect to . This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D'Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante, by providing a deeper insigt on the ideal-theoretic structure of bi-amalgamations.
Paper Structure (5 sections, 20 theorems, 28 equations)

This paper contains 5 sections, 20 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Noetherian bi-amalgamations
  4. Prüfer bi-amalgamations
  5. Gaussian bi-amalgamations

Key Result

Lemma 2.2

If $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are two ideals of $A$ containing $\mathfrak{i}_0$ such that $\mathfrak{a}_1\bowtie^{f,g}(\mathfrak{b},\mathfrak{c})\subseteq \mathfrak{a}_2\bowtie^{f,g}(\mathfrak{b},\mathfrak{c})$, then $\mathfrak{a}_1\subseteq \mathfrak{a}_2$.

Theorems & Definitions (45)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 35 more