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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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2 Answers 2

Since I don't have enough reputation, I have to write this as an answer: Do you mean isomorphic as rings (without unit) or modules or groups or ...? If you mean morphism of modules, consider principal ideals.

The question depends heavily on the category (rings, rngs, modules, groups, etc.) you are working in.

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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.

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