@Youngsu If the height of an arbitrary ideal behaves so badly when localize and take factor rings, how can you conclude in your answer to the original problem that $\text{height }I\le\text{height }I'+ 1$? In fact you got the following: $\text{height }(I+q)/q\le 1$ in $R/q$. Is this enough to prove that $\text{height }I\le\text{height }I'+ 1$? Or maybe I didn't get your reasoning?

@Lao-tzu Your comment is wrong: $x^5=x^2x^3\in(x^2,x^3)^2$, and $x^5\notin x^4k[x^2,x^3]$. As it is said, $\mathfrak m^2=(x^5)$.