Amalgamated algebras along an ideal. (English) Zbl 1177.13043

D’Anna, Marco; Finocchiaro, Carmelo Antonio; Fontana, Marco

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].

MSC:

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

Keywords:

Nagata’s idealization; pullback; \(D+M\) construction; amalgamated duplication

Citations:

Zbl 0003.38701; Zbl 1126.13002; Zbl 1143.13002; Zbl 0363.13002; JFM 58.0137.02