No element of degree $0$ (constant $= a_0 X^0$) in $\mathbb{Z}_4/(2X) \mathbb{Z}_4$

Question

Consider $\mathbb{Z}_4[X]/(2X) \mathbb{Z}_4[X]$. Then we wish to show/verify that the residue class $X$ does not contain an element of degree $0$.

In the previous exercise the book asked that if we have a monic polynomial in $R[X]$ then division by this polynomial causes the residue classes to be of lesser degree. Now of course $2X^1$ is not monic, so that does not mean it has to have a constant residue class($X^0$), but in fact we need to show that there are NONE. Why is this the case?

Answer 1

$X = a X^0$ in $R\!\iff\! X = a X^0\! + (2X) f(X)\,$ in $\,\Bbb Z_4[X]$ $\,\Rightarrow\, a = 0\, $ in $\,\Bbb Z_4,\,$ by evaluation at $\,X = 0$