A constructive proof of Orzech's theorem

TL;DR

This paper provides a constructive proof of Orzech's theorem for finitely generated modules over commutative rings, utilizing the Cayley-Hamilton theorem to establish the result.

Contribution

It offers a new constructive proof of Orzech's theorem, which was previously proven non-constructively, using the Cayley-Hamilton theorem.

Findings

Proves that surjective homomorphisms from submodules to the module are isomorphisms.

Uses Cayley-Hamilton theorem in a novel way for module theory.

Provides a constructive approach to a classical algebraic result.

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.