Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Commutative ring theory: Regular rings

Commutative rings I

We introduce a new general construction, denoted by R ⋈ E, called the amalgamated duplication of a ring R along an R-module E, that we assume to be an ideal in some overring of R. (Note that, when E2 = 0, R ⋈ E coincides with the Nagata's idealization R ⋉ E.). After discussing the main properties of the amalgamated duplication R ⋈ E in relation with pullback-type constructions, we restrict our investigation to the study of R ⋈ E when E is an ideal of R. Special attention is devoted to the ideal-theoretic properties of R ⋈ E and to the topological structure of its prime spectrum.

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An amalgamated duplication of a ring along an ideal: the basic properties

A construction of Gorenstein rings

We introduce a new general construction, denoted by $R\JoinE$, called the amalgamated duplication of a ring $R$ along an $R$--module $E$, that we assume to be an ideal in some overring of $R$. (Note that, when $E^2 =0$, $R\JoinE$ coincides with the Nagata's idealization $R\ltimes E$.) 
After discussing the main properties of the amalgamated duplication $R\JoinE$ in relation with pullback--type constructions, we restrict our investigation to the study of $R\JoinE$ when $E$ is an ideal of $R$. 
Special attention is devoted to the ideal-theoretic properties of $R\JoinE$ and to the topological structure of its prime spectrum.

After recalling briefly the main properties of the amalgamated duplication of a ring R along an ideal I, denoted by $R\bowtie I$, see M. D’Anna and M. Fontana, to appear in J. Algebra Appl., we restrict our attention to the study of the properties of $R\bowtie I$, when I is a multiplicative canonical ideal of R, see W. J. Heinzer, J. A. Huckaba and I. J. Papick, Comm. Algebra. In particular, we study when every regular fractional ideal of $R\bowtie I$ is divisorial.

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The amalgamated duplication of a ring along a multiplicative-canonical ideal

http://www.mat.uniroma3.it/users/fontana/documents/da-fi-fo-arxiv-JPAA.pdf

Marco D'Anna

Papers

An amalgamated duplication of a ring along an ideal: the basic properties

A construction of Gorenstein rings

An amalgamated duplication of a ring along an ideal: the basic properties

The amalgamated duplication of a ring along a multiplicative-canonical ideal

Properties of chains of prime ideals in an amalgamated algebra along an ideal