# Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.

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If you mean a proper ideal, i.e. an ideal which is not the whole ring, then no. To see this, note that $F[X]$ has a unit while no proper ideal does, and the image of a unit under an isomorphism is a unit.