Amalgamated algebras along an ideal. (English) Zbl 1177.13043

Fontana, Marco (ed.) et al., Commutative algebra and its applications. Proceedings of the fifth international Fez conference on commutative algebra and applications, Fez, Morocco, June 23–28, 2009. Berlin: Walter de Gruyter (ISBN 978-3-11-020746-0/hbk). 155-172 (2009).
Summary: 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 [Bull. Am. Math. Soc. 38, 85–88 (1932; Zbl 0003.38701; JFM 58.0137.02))] and provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by the first and third author [J. Algebra Appl. 6, No. 3, 443–459 (2007; Zbl 1126.13002); Ark. Mat. 45, No. 2, 241–252 (2007; Zbl 1143.13002)], and other classical constructions such as the \(A+XB[X]\) and \(A+XB[[X]]\) constructions, the CPI-extensions of M. B. Boisen jun. and P. B. Sheldon [Can. J. Math. 29, 722–737 (1977; Zbl 0363.13002)], the \(D+M\) constructions and the Nagata’s idealization.
For the entire collection see [Zbl 1175.13001].


13E05 Commutative Noetherian rings and modules
13B99 Commutative ring extensions and related topics
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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