I am currently working my way through Bosch, Algebraic Geometry and Commutative Algebra. I want to solve Exercise 1, Chapter 2.4:
Consider the polynomial ring $R[X]$ in one variable over a not necessarily Noetherian ring. Show $\dim R+1\leq \dim R[X]\leq 2⋅\dim R+1$. Hint: Let $p_1\subset p_2\subset R[X]$ be two different prime ideals in $R[X]$ restricting to the same prime ideal $p\subset R$. Deduce $pR[X]=p_1$.
The first inequality is no problem. But I have no idea how to prove the hint. May anyone give me a hint?