Dear @Youngsu : Could you please tell me when the equalities occur?

Dear Thomas Kahle, in the last line of your answer, you concluded that $\text{ht}(I)=\text{ht}(I')+1$, but we have to prove that $\text{ht}(I)\le\text{ht}(I')+1$ and why do we get $\text{ht}(IR_{(0)})=1$ ? I am so sorry if you feel my question is stupid. Thank you very much for your answer.

Sorry, but in your answer you has just dealt with only prime ideal. My question is about a general ideal.