Amalgamated algebras along an ideal

TL;DR

This paper introduces a new ring construction called the amalgamation of A with B along an ideal, unifying and generalizing several classical ring constructions and providing a systematic framework for their study.

Contribution

It systematically studies the amalgamation of rings along an ideal, extending classical constructions and unifying various ring-theoretic frameworks.

Findings

Provides a new general construction unifying classical ring constructions.

Establishes foundational properties of the amalgamation along an ideal.

Connects the new construction to existing algebraic frameworks.

Abstract

Let $f:A \to B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we initiate a systematic study of a new ring construction called the "amalgamation of $A$ with $B$ along $J$ with respect to $f$". This construction finds its roots in a paper by J.L. Dorroh appeared in 1932 and provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D'Anna and Fontana in 2007, and other classical constructions such as the $A+ XB[X]$ and $A+ XB[[X]]$ constructions, the CPI-extensions of Boisen and Sheldon, the $D+M$ constructions and the Nagata's idealization.