On the square of an isosimple module

Ran Yan; Xiaolei Zhang

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arXiv · 2610.26905

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 $S^2 := S \oplus S$ 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 $\mathrm{End}_R(S)$ 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 $E = \mathrm{End}_R(S)$ is either commutative Noetherian or right semihereditary, then every finite power of $S$ is isoantian.

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Ran Yan, Xiaolei Zhang. 2026-10-17. On the square of an isosimple module. https://arxiv.org/abs/2610.26905

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