I'm confused about a definition and I've searched through books and the internet, but I couldn't find anything. I hope someone knowledgeable about the topic can explain it to me.

To begin, let's consider the following definition:

Let $R$ be a commutative ring. We say that $f \in R[x_1, \cdots, x_n]$ is a coordinate in $R[x_1, \cdots, x_n]$ if there exists an $R$-automorphism $\phi$ of $R[x_1, \cdots, x_n]$ such that $\phi(f) = x_1$.

I found this definition here.

Now, I understand that a morphism of $R$-modules is also called $R$-linear, and a morphism of rings is simply called a ring homomorphism. However, my question here is: what does the author mean by an $R$-automorphism of $R[x_1, \cdots, x_n]$? Is it a ring automorphism, a module automorphism, or something else?


In addition to the aforementioned points, another definition is provided on page 2 of the same paper. It states:

$f\in R[x_1,\cdots, x_n] $ is a coordinate in $R[x_1,\cdots, x_n] $ if there exist $f_2,\cdots, f_n$ such that $R[f, f_2,\cdots, f_n] =R[x_1,\cdots, x_n]$.

Thus, one might wonder how we can demonstrate the equivalence of these two definitions.

  • $\begingroup$ $R[x_1,\ldots,x_n]$ has both a structure of an $R$-module, and a ring structure. It is asking $f$ to be an $R$-module automorphism of $R[x_1,\ldots,x_n]$. $\endgroup$ May 8 at 19:37
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    $\begingroup$ Not just that; $R$-linear means it is an $R$-module homomorphism. "Automorphism" means that it is an isomorphism from the object to itself. So $R$-automorphism means that it is an invertible $R$-module homomorphism (i.e., an $R$-module isomorphism) from the module to itself. $\endgroup$ May 8 at 19:43
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    $\begingroup$ Okay great it's done. $\endgroup$ May 8 at 19:46
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    $\begingroup$ To me the natural way to interpret this is that $\phi$ should be an $R$-algebra automorphism, in the same way that one talks about "$K$-automorphisms" in Galois theory for example. If it just meant $R$-module homomorphism, then it seems that every nonzero polynomial over a field would be a coordinate, which I don't think is true. But I don't really know much about coordinates. @ArturoMagidin, sorry to bother you - am I talking nonsense? $\endgroup$ May 8 at 20:55
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    $\begingroup$ @IzaakvanDongen Actually you are probably right. Note that the "definition" is literally just a parenthetical comment in the paper linked. $\endgroup$ May 8 at 20:59

1 Answer 1


$R$-automorphism here means automorphism of $R$-algebras. So, it's a ring automorphism that preserves the $R$-algebra structure as well: $\phi(r f) = r\phi(f)$ for $r \in R$ and $f \in R[x_1, \dots x_n]$. Differently said, it's a ring automorphism that is also an $R$-module automorphism.

The canonical reference for this is, at least in the context of the paper you link to, Van den Essen, Polynomial Automorphisms and the Jacobian Conjecture (2000).


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