# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 6, 2022

## ON ALMOST QUASI-COHERENT RINGS AND ALMOST VON NEUMANN RINGS

Haitham EL ALAOUI, Mourad EL MAALMI, and Hakima MOUANIS

Faculty of Sciences Dhar Al Mahraz, Laboratory of geometric and arithmetic algebra, Fez, Morocco.

### Abstract

Let R be a commutative ring with identity. We call the ring $R$ to be a almost quasi-coherent rings 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 $\bigcap\limits_{i=1}^p Ra_{i}^{m}$ and $Ann_{R}(a^{m})$ are finitely generated, and to be a 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.