Zero intersection of two ideals in the ring $R=\mathbb{Z}_{4}\left[ x\right] /\left\langle x^{n}-1\right\rangle .$

Question

Let $n$ be an odd integer and $f,g,h$ be monic polynomials over $\mathbb{Z}_{4}$ such that $x^{n}-1=fgh.$ Also let $\left\langle fh,2fg\right\rangle $ and $\left\langle g^{\ast }h^{\ast },2f^{\ast }g^{\ast }\right\rangle $ (where $"^{\ast }"$ represents the reciprocal of the polynomials) be two ideals of the quotient ring $R=\mathbb{Z}_{4}\left[ x\right] /\left\langle x^{n}-1\right\rangle .$ I am trying to determine under which conditions the intersection $\left\langle fh,2fg\right\rangle \cap \left\langle g^{\ast }h^{\ast },2f^{\ast }g^{\ast }\right\rangle $ is $\left\{ 0\right\} .$ To do this, I know that $lcm\left( fh,g^{\ast }h^{\ast }\right) |x^{n}-1$ and $lcm\left( fg,f^{\ast }g^{\ast }\right) |x^{n}-1,$ but I can't go further. Thanks in advance!