On almost quasi-coherent rings and almost von Neumann rings
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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.
Abstract
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.
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