Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring with identity. We call the ring $R$ to be an almost quasi-coherent ring if for any finite set of elements $a_{1},\dots,a_{p}$ and $a$ of $R$, there exists a positive integer $m$ such that the ideals $\bigcap_{i=1}^p Ra_{i}^{m}$ and $Ann_{R}(a^{m})$ are finitely generated, and to be almost von Neumann regular rings if for any two elements $a$ and $b$ in $R$, there exists a positive integer $n$ such that the ideal $(a^{n}, b^{n})$ is generated by an idempotent element. This paper establishes necessary and sufficient conditions for the Nagata's idealization and the amalgamated algebra to inherit these notions. Our results allow us to construct original examples of rings satisfying the above-mentioned properties.

Keywords: Almost quasi-coherent rings, almost von Neumann regular rings, trivial rings extension, amalgamated algebra along an ideal

MSC numbers: 15A03, 13A15, 13B25, 13D05, 13F05