A short proof that the free associative algebra is Hopfian
Alexei Kanel-Belov, Louis Rowen, and Jie-Tai Yu
TL;DR
This paper presents a concise proof demonstrating that free associative algebras are Hopfian, meaning all surjective endomorphisms are automorphisms, which simplifies previous solutions related to the Jacobian conjecture.
Contribution
It provides a simplified proof that free associative algebras are Hopfian, extending to various algebra classes and streamlining prior complex arguments.
Findings
Free associative algebra is Hopfian.
Simplifies the Dicks-Lewin solution of the Jacobian conjecture.
Applicable to various algebra classes.
Abstract
A short proof is given of the fact that various classes of algebras including the free associative algebra are Hopfian, i.e., every epimorphism is an automorphism. This further simplifies the Dicks-Lewin solution of the Jacobian conjecture for the free associative algebra.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra · Polynomial and algebraic computation