Perspectivity and the perspective-Schröder-Bernstein Property

Abstract

In this paper, we study various aspects of perspectivity in modules. We prove that the following classes of modules satisfy the perspective-Schröder-Bernstein property: modules with transitive perspectivity, quasi-continuous modules, Harada (and hence discrete) modules and quasi-discrete modules with the (finite) exchange property. Furthermore, we prove that for a semiregular ring, the Schröder-Bernstein property implies the perspective-Schröder-Bernstein property. We prove that for A,B ⊆ ^⊕ M, if all complements of A are perspective with B, then all complements of B are perspective with A. We also provide new characterizations of weakly perspective modules and modules in which perspectivity is transitive. Some applications of these results are given. We finally prove that perspectivity is transitive in a ring R if and only if every special clean element of R is perspective.

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BibTeX RIS

BibTeX

@misc{khurana2026perspectivity,
  title = {{Perspectivity and the perspective-Schröder-Bernstein Property}},
  author = {Dinesh Khurana and Shubham Mittal},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39491},
  eprint = {2609.39491},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39491},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Perspectivity and the perspective-Schröder-Bernstein Property
AU  - Dinesh Khurana
AU  - Shubham Mittal
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39491
UR  - https://arxiv.org/abs/2609.39491
AB  - In this paper, we study various aspects of perspectivity in modules. We prove that the following classes of modules satisfy the perspective-Schröder-Bernstein property: modules with transitive perspectivity, quasi-continuous modules, Harada (and hence discrete) modules and quasi-discrete modules with the (finite) exchange property. Furthermore, we prove that for a semiregular ring, the Schröder-Bernstein property implies the perspective-Schröder-Bernstein property. We prove that for A,B ⊆ ^⊕ M, if all complements of A are perspective with B, then all complements of B are perspective with A. We also provide new characterizations of weakly perspective modules and modules in which perspectivity is transitive. Some applications of these results are given. We finally prove that perspectivity is transitive in a ring R if and only if every special clean element of R is perspective.
ER  -

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