Transitivity of Perspectivity

Algebras and Representation Theory (2021)Cite this article

Abstract

We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in \(\mathbb {M}_{2}(R)\) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective.

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Acknowledgements

We thank Xavier Mary for comments on our paper, as well as sending us an alternative proof of Theorem 2.5.

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Affiliations

  1. Department of Mathematics, Panjab University, Chandigarh, 160014, India

    Dinesh Khurana

  2. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Pace P. Nielsen

Authors
  1. Dinesh Khurana
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  2. Pace P. Nielsen
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Correspondence to Pace P. Nielsen.

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Khurana, D., Nielsen, P.P. Transitivity of Perspectivity. Algebr Represent Theor (2021). https://doi.org/10.1007/s10468-020-10020-y

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Keywords

  • Dedekind-finite
  • Quasi-continuous modules
  • Stable range one
  • (Transitivity of) perspectivity
  • Von Neumann regular rings

Mathematics Subject Classification (2010)

  • Primary 16D70
  • Secondary 16E50
  • 16U99