On the prime spectrum of the amalgamations

Abstract

Let $f:R\to S$ be a ring homomorphism and $J$ be an ideal of $S$. Then the subring $R{\bowtie }^{f}J:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we will deepen the study of the prime spectrum of the amalgamations and characterize when $R{\bowtie }^{f}J$ is a pm-ring. We also investigate the transfer of compactly packed and properly zipped property on amalgamated constructions.

Example 3.2 and Remark 3.3 are submitted in "Journal of Algebra and its Applications" (at Feb 17, 2024), as a counterexample for the main result of a paper (reference [15]) that noted in the current paper. It is also shown that, the result is valid under an additional assumption