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Properties of chains of prime ideals in an amalgamated algebra along an ideal. (English) Zbl 1191.13006

Let \(A\) and \(B\) be commutative rings with unity, let \(J\) be an ideal of \(B\) and let \(f:A \to B\) be a ring homomorphism. Consider the following subring of \(A \times B\): \({A\bowtie^fJ}:= \{(a, f(a) +j)\mid a \in A, j \in J\}\) called the amalgamation of \(A\) with \(B\) along \(J\) with respect to \(f\).
In this paper, the authors study this amalgamation, which is a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, which was 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 \(A+ XB[X]\), the \(A+ XB[[X]]\) and the \(D+M\) constructions). In particular, the authors completely described the prime spectrum of the amalgamated duplication and they give bounds for its Krull dimension.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13B99 Commutative ring extensions and related topics
14A05 Relevant commutative algebra
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References:

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