Does $Z[x]/(3,p(x))$ equal to $Z_3[x]/(p(x))$ for all integer coefficient polynomials?
Justifying $\,\Bbb{Z}[i]/\langle 2\rangle\cong \Bbb{Z}[X]/\langle X^2+1,2\rangle\cong \Bbb{F}_2[X]/\langle X^2+1\rangle$