# Homomorphism or derivation conserving irreducibility

Let $$R$$ be a integral domain and $$\phi$$ be an automorphism of $$R$$. For a given element $$x \in R$$, we consider a sequence $$(\phi^n(x))_{n=0}^{\infty}$$.

I wonder if there is any related theory to determine when $$\phi^n(x)$$ is irreducible for all $$n \in \mathbb Z_{\geq 0}$$. It depends on $$x$$ and $$\phi$$ of course.

More precisely, I want to know about a specific case, not so general: $$R=k[x_1,\ldots,x_n]$$, where $$k$$ is a field.

Any suggestions and comments are welcome.

--

I want to know about the derivation, so I edit the question.

Let $$R$$ be an algebra not just a ring. In fact I wonder only when $$R$$ is a polynomial ring, which is an $$k$$-algebra, and $$\phi$$ is not a ring homomorphism, but just a derivation. (Which is an $$k$$-module homomorphism).

Or more generally, $$R$$:integral domain, and $$\phi$$ is a derivation (satisfying Leibniz rule and group homomorphism).

• A derivation is not an automorphism. – abx Jun 8 at 6:15
• abx // Right. that's my fault. I'll edit. – LWW Jun 8 at 6:18
• An automorphism preserves and reflects irreducibility, so you have to change that too – Max Jun 9 at 10:52
• Max // yeah I know, so I changed the question for a derivation, not an automorphism. – LWW Jun 9 at 11:24