Q:

Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it is possible to construct a ring $R$ such that $p(x)$ has a root in $R$.

Prove or disprove: Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it
  is possible to construct a ring $R$ such that $p(x)$ has a root in
  $R$.

For this exercise I think about complex numbers. Since each polynomial has complex root we can take complex numbers ring. Is it a right solution or I am missing something?

A:

Just follow the usual construction and embed $\mathbb{Z}_6 \hookrightarrow \mathbb{Z}_6 [X]/(p(x))$ (provided that $p$ is not constant).

