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Characterizations of Gelfand Rings Specially Clean Rings and their Dual Rings

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Abstract

In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.

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Acknowledgements

The authors would like to give sincere thanks to Professors Pierre Deligne and François Couchot for very valuable correspondences during the writing of the present paper. We would also like to give heartfelt thanks to the referees for very careful reading of the paper and for their very valuable comments and suggestions which improved the paper.

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  1. Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran

    Mohsen Aghajani & Abolfazl Tarizadeh

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  1. Mohsen Aghajani
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  2. Abolfazl Tarizadeh
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Correspondence to Abolfazl Tarizadeh.

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Aghajani, M., Tarizadeh, A. Characterizations of Gelfand Rings Specially Clean Rings and their Dual Rings. Results Math 75, 125 (2020). https://doi.org/10.1007/s00025-020-01252-x

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  • DOI: https://doi.org/10.1007/s00025-020-01252-x

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