Let $A$ be a commutative ring and let $f: A \rightarrow A$ an surjective homomorphism, let $a$ be a ideal of $A$ then if $f(a)\subseteq a$ then it's $f$ is the identity map, or not necessary.
2026-04-14 19:17:55.1776194275
Is this automorphism the identity map
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No. Let $f:k[x,y]\to k[x,y]$ switch $x$ and $y$. Then ideals like $(x,y)$ and $(xy)$ are preserved, but $f$ is not the identity.