## 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

### Citations:

Zbl 1126.13002; Zbl 1143.13002
Full Text:

### References:

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