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Keywords:
retract; locally nilpotent derivation; kernel; Zariski's cancellation problem
retract; locally nilpotent derivation; kernel; Zariski's cancellation problem
Summary:
Let $k$ be a field of characteristic zero and $B$ a $k$-domain. Let $R$ be a retract of $B$ being the kernel of a locally nilpotent derivation of $B$. We show that if $B=R\oplus I$ for some principal ideal $I$ (in particular, if $B$ is a UFD), then $B= R^{[1]}$, i.e., $B$ is a polynomial algebra over $R$ in one variable. It is natural to ask that, if a retract $R$ of a $k$-UFD $B$ is the kernel of two commuting locally nilpotent derivations of $B$, then does it follow that $B\cong R^{[2]}$? We give a negative answer to this question. The interest in retracts comes from the fact that they are closely related to Zariski's cancellation problem and the Jacobian conjecture in affine algebraic geometry.
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