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arxiv: 0807.1142 · v1 · submitted 2008-07-07 · 🧮 math.RA

Test elements, retracts and automorphic orbits

classification 🧮 math.RA
keywords elementautomorphictestalgebraassociativeautomorphismbelongcharacteristic
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Let $A_2$ be a free associative or polynomial algebra of rank two over a field $K$ of characteristic zero. Based on the degree estimate of Makar-Limanov and J.-T.Yu, we prove: 1) An element $p \in A_2$ is a test element if $p$ does not belong to any proper retract of $A_2$; 2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $A_2$ is an automorphism.

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