Algebraic and topological properties of an amalgamated algebra along an ideal

Abstract

Let $f:A\to B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$, a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+XB[X]$, the $A+XB[[X]]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamation and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen-Macaulay ring or a Gorenstein ring.

The present version of the manuscript differs from the previous one for an appendix where we note that Proposition 4.1(2) and Theorem 4.4 hold under the assumption, not explicitly declared, that B=f(A)+J. In the same appendix, we provide the exact value of $\mathrm{embdim}(A{\bowtie }^{f}J)$, under the hypothesis that J is finitely generated as an ideal of the ring f(A)+J. Finally, we deleted Example 4.6

Cited by in corpus (1)

• Cohen-Macaulay and Gorenstein properties under the amalgamated construction