Let $R$ be a Noetherian integral domain of finite Krull-dimension and $0 \neq x \in R$ a non-unit. Do we have $\dim(R/x) = \dim(R) -1$ in general?

If this is wrong, does it change something if we further assume that $R$ is positively graded, finitely generated by homogeneous elements of degree one over $R_0$, which is Artinian local, and $x$ is homogeneous of degree one?

Context: This dimension formula is used here.