I'm struggling with the following problem. Any help is greatly appreciated!
Let $R$ be a local integral domain with the unique maximal ideal $\mathfrak m$. Let $g(x), h(x)$ be monic polynomials in $R[x]$ such that $(R/\mathfrak m)[x]$ is generated (as ideal) by the reductions of $g(x)$ and $h(x)$ modulo $\mathfrak m$, i.e., the polynomials obtained by reducing the coefficients of $g(x)$ and $h(x)$ modulo $\mathfrak m$. Show that $R[x]$ is generated (as ideal) by $g(x)$ and $h(x)$.