A Survey on Algebraic and Homological Properties of Amalgamated Algebras of Commutative Rings

Maryam Salimi (), Elham Tavasoli () and Siamak Yassemi ()
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Maryam Salimi: Islamic Azad University, Department of Mathematics, East Tehran Branch
Elham Tavasoli: Islamic Azad University, Department of Mathematics, East Tehran Branch
Siamak Yassemi: University of Tehran, School of Mathematics, Statistics and Computer Science, College of Science

A chapter in Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 2023, pp 383-404 from Springer

Abstract: Abstract Let A and B be commutative rings with identity and J be an ideal of B, and let f : A → B be a ring homomorphism. The amalgamation of A with B along J with respect to f denoted by A⋈fJ was introduced by D’Anna, Finocchiaro, and Fontana in 2010. This construction is a generalization of the amalgamated duplication of a ring along an ideal introduced by D’Anna and Fontana in 2007. Several classical constructions such as A + XB[X], A + XB[[X]], and D + M constructions can be studied as particular cases of the amalgamation. Also, other classical constructions such as Nagata’s idealizations and CPI extensions are strictly related to amalgamation. Since the amalgamation can be studied in the frame of pullback constructions, various properties of A⋈fJ in connection with the properties of A, J, and f are given by several authors. The purpose of this paper is to survey recent algebraic and homological properties of the ring A⋈fJ.

Date: 2023
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DOI: 10.1007/978-3-031-28847-0_20

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