Justification of an isomorphism between $\mathbb{Z}[t]/(t,3)$ and $\mathbb{Z}/3$ I am wondering if the following is true:
$$\mathbb{Z}[t]/(t,3)\cong\mathbb{Z}/3.$$
I can tell that the ideal $(t,3)$ is maximal. In contrast, the ideal $(t^2,2t)$ is not even irreducible. So it's less likely that we can simplify $\mathbb{Z}[t]/(t^2,2t)$ further.
If the aforementioned isomorphism is true, that means $\mathbb{Z}[t]/(t,3)\cong(\mathbb{Z}[t]/t)/(3)\cong\mathbb{Z}/3$, where the $(3)$ is an ideal in $\mathbb{Z}[t]/t$. My questions are:
1) Is this argument legitimate?
2) If it is true, is there any theorem justifying the aforementioned isomorphism?

It's legitimate and a consequence of the isomorphism theorems. 
