Journal ArticleDOI
New Algebraic Properties of an Amalgamated Algebra Along an Ideal
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TLDR
In this paper, the authors studied the prime spectrum of 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).Abstract:
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.read more
Citations
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Journal ArticleDOI
Cohen-macaulay and gorenstein properties under the amalgamated construction
TL;DR: In this article, the Cohen-Macaulay and (quasi-)Gorenstein properties on the ring A⋈fJ were investigated, among other things, for ring homomorphisms.
Journal ArticleDOI
Prüfer conditions in amalgamated algebras
TL;DR: In this paper, the transfer of Prufer, Gaussian, and arithmetical conditions in amalgamated algebras has been studied, and an answer to a question posed by Chatterjee is given.
Journal ArticleDOI
On n-absorbing and strongly n-absorbing ideals of amalgamation
TL;DR: In this article, the authors define an n-absorbing ideal I of a commutative ring with 1 ≤ 0 and n ≥ 1, where n is a positive integer.
Journal ArticleDOI
Commutative rings and modules that are Nil*-coherent or special Nil*-coherent
TL;DR: In this article, the authors define and study Nil∗-coherent modules and special NCC modules over any ring and establish the basic properties of these modules and their properties.
Book ChapterDOI
Ring and Semigroup Constructions
TL;DR: This paper presents a survey on some ring constructions, recently introduced and studied, and shows how to produce some analogous semigroup constructions; and describes how to translate at semigroup level some ring properties of these constructions.
References
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Book
Commutative Algebra: with a View Toward Algebraic Geometry
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Book
Introduction to Commutative Algebra
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Book
Cohen-Macaulay rings
Winfried Bruns,H. Jürgen Herzog +1 more
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Book
Integral closure of ideals, rings, and modules
Craig Huneke,Irena Swanson +1 more
TL;DR: In this paper, the authors define the integral closure of rings and define a table of basic properties including separation, separationability, separation of rings, and normal homomorphisms, and the Briancon-Skoda theorem.