For which ideal $I$ of $\Bbb Z[t]$ is $\mathbb{Z}[t]/I\cong\Bbb Z_{11}$?

Question

Maybe for $I=(11,t-1)$ but i don't know how to prove it or if it is even right.

Answer 1

We want to kill $11$ and $t$ and so the natural choice is $I=(11,t)$. This leads to the homomorphism $\phi: \mathbb{Z}[t] \to \mathbb{Z}_{11}$ given by $\phi(p(t)) = p(0) \bmod 11$. It remains to prove that $\phi$ is surjective with kernel $I$.

Answer 2

Use Third Isomorphism theorem $$\frac{\Bbb Z[t]}{(11,t-1)}\simeq \frac{\Bbb Z[t]/(t-1)}{(11,t-1)/(t-1)}\simeq \frac{\Bbb Z}{(11)}\simeq \Bbb Z_{11}.$$