I'm studying some commutative algebra, and i've stepped into an exercise but i don't know if what i am doing is correct, so just wanted a proof check. The exercise is as it follows
Let $A = \mathbb{R}[X,Y,Z]$ and $p = <X-3,Y-3/7>$. Determine the Krull dimension of the localization $A_p$.
Considering that $$ 0 \subset \, <X- a_1> \, \subset \, <X- a_1, Y-a_2> \, \subset \, <X- a_1, Y-a_2, Z-a_3> $$ is a saturated ascending chain of prime ideals, and that the prime ideals of $A_p$ are one-to-one with prime ideal in $A$ contained in $p$, my natural guess is that $\dim A_p = \operatorname{ht} P - 1 = 1$. Is that correct?