Prime elements and irreducible elements of quotient rings of polynomial ring We know that $\mathbb{Z}[x]$ is a euclidean domain hence prime elements and irreducible elements are the same. Let $f(x)\in \mathbb{Z}[x]$ and $R=\mathbb{Z}[x]/(f(x))$. Let $\phi: \mathbb{Z}[x]\longrightarrow R$, $\phi:g(x)\longmapsto g(x)+(f(x))$. 

(1) If $g(x)$ is prime in $\mathbb{Z}[x]$, will $\phi(g(x))$ be prime in $R$?
(2) If $g(x)$ is irreducible in $\mathbb{Z}[x]$, will $\phi(g(x))$ be irreducible in $R$? 
(3) Can we put any conditions on $f(x)$ such that (1), (2) are true?
(4) What will happen in the case $\mathbb{Z}[x_1,\dots,x_n]/(f(x_1,\dots,x_n))$?
(5) How about $\mathbb{Q}[x]/(f(x))$ and $\mathbb{Q}[x_1,\dots,x_n]/(f(x_1,\dots,x_n))$?

Are there any chapters or theorems about these questions? I am confused on these questions.

There are some trivial cases for (1) and (2), such as $g$ is mapped to zero, or to a unit.
Among the non-trivial examples, $2$ is prime and irreducible in $\mathbb{Z}[x]$, but not in the Euclidean domain $\mathbb{Z}[i] \cong \mathbb{Z}[x]/(x^{2} + 1)$.
A variation on this yields examples for (4).
