Bulletin of the Korean Mathematical Society (대한수학회보)

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ON ALMOST QUASI-COHERENT RINGS AND ALMOST VON NEUMANN RINGS

  • El Alaoui, Haitham (Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University) ;
  • El Maalmi, Mourad (Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University) ;
  • Mouanis, Hakima (Faculty of Sciences Dhar El Mahraz Sidi Mohamed Ben Abdellah University)
  • Received : 2021.09.12
  • Accepted : 2021.11.12
  • Published : 2022.09.30
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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 α1, …, αp and a of R, there exists a positive integer m such that the ideals $\bigcap{_{i=1}^{p}}\;R{\alpha}^m_i$ and AnnR(αm) 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 (αn, bn) 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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Acknowledgement

The authors would like to express their sincere thanks to the referee for his/her helpful suggestions and comments.

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