How does one find the Krull dimension of a composite ring?

Question

For example, if the ring is $\mathbb{Z} + X \mathbb{Q}[X]$. Is the dimension $1$?

Answer 1

There are no formulas for the Krull dimension of the composite rings $A+XB[X]$, instead I can record this result:

Let $A\subset B$ be integral domains, $S=A\setminus\{0\}$, and $R=A+XB[X]$. Then $$\max(\dim A+\dim S^{-1}B[X],\dim B[X])\le\dim R\le\dim A+\dim B[X].$$ When $Q(A)\subseteq B$ we have $\dim R=\dim A+\dim B[X].$

In your example $\dim(\mathbb Z+X\mathbb Q[X])=2.$