Let $ R = \mathbb{ F }_2[x]/(x^2) $. Determine all the ideals in $ R $.

Question

(3) Let $ R = \mathbb{ F }_2[x]/(x^2) $. Determine all the ideals in $ R $.

$$ R := \frac{ \mathbb{ F }_2[x] }{ (x^2) },$$

the quotient of the polynomial ring $ \mathbb{ F }_2[x] $ (read "f two $ x $") over $ \mathbb{ F }_2 $ (i.e., polynomials whose coefficients belong the field of two elements, $ \mathbb{ F }_2 $, by the principal ideal ($ x^2 $).

Answer 1

$R=\{0,1,x,x+1\}$.

Thus $(0),(1)=R, (x)$ and $(x+1)$ are the only candidates. (Of course, $(x,x+1)=R$).

That leaves two possible nontrivial ideals: $(x)$ and $(x+1)$. But, $x\cdot (x+1)=x^2+x=x$. So $(x+1)=R$.

That leaves only one ideal: $(x)$.

Alternatively, $(x)$ is the only (nontrivial) ideal in $\mathbb F_2[x]$ containing $(x^2)$.