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On the square of an isosimple module

  • Ran Yan
  • Xiaolei Zhang
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Abstract

Let S be an isosimple module, that is, a nonzero module isomorphic to every nonzero submodule of itself. In this paper, we investigate if the square S2 := S ⊕ S is isoantian when S is an isosimple module. If the base ring is commutative, then the answer is yes. However, for noncommutative rings this is no longer true even when EndR(S) is a discrete valuation ring. We also give some positive results coming from quasi-projectivity and Morita equivalence. Finally, we consider when every finite powers of an isosimple module are isoantian provided it holds for the square. Our related result says that if S2 is isoantian and E = EndR(S) is either commutative Noetherian or right semihereditary, then every finite power of S is isoantian.

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Cite

BibTeXRIS
BibTeX
@misc{yan2026square,
  title = {{On the square of an isosimple module}},
  author = {Ran Yan and Xiaolei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26905},
  eprint = {2610.26905},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26905},
}
RIS
TY  - UNPB
TI  - On the square of an isosimple module
AU  - Ran Yan
AU  - Xiaolei Zhang
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.26905
UR  - https://arxiv.org/abs/2610.26905
AB  - Let S be an isosimple module, that is, a nonzero module isomorphic to every nonzero submodule of itself. In this paper, we investigate if the square S^2 := S ⊕ S is isoantian when S is an isosimple module. If the base ring is commutative, then the answer is yes. However, for noncommutative rings this is no longer true even when End_R(S) is a discrete valuation ring. We also give some positive results coming from quasi-projectivity and Morita equivalence. Finally, we consider when every finite powers of an isosimple module are isoantian provided it holds for the square. Our related result says that if S^2 is isoantian and E = End_R(S) is either commutative Noetherian or right semihereditary, then every finite power of S is isoantian.
ER  -