A constructive proof of Orzech's theorem

Darij Grinberg

A constructive proof of Orzech's theorem

Commutative Algebra 2026-04-20 v2 Rings and Algebras

Authors: Darij Grinberg

Abstract

Let $A$ be a commutative ring with unity, and $M$ a finitely generated $A$ -module. In 1971, Morris Orzech showed that any surjective $A$ -module homomorphism from a submodule of $M$ to $M$ must be an isomorphism. We give a constructive proof of this fact using the Cayley--Hamilton theorem.

Cite

@article{arxiv.2604.13911,
  title  = {A constructive proof of Orzech's theorem},
  author = {Darij Grinberg},
  journal= {arXiv preprint arXiv:2604.13911},
  year   = {2026}
}

Comments

9 pages. Note written in 2015, mildly edited and with open questions added at the end. Posted mainly because I keep citing it. v2 fixes a mistake in the abstract and adds missing references

A constructive proof of Orzech's theorem

Abstract

Cite

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