Dimension of a quotient ring

Question

What is the Krull dimension of $B=A[x,y,z]/\langle xy + 1, z + 1\rangle$, given $A$ is a Noetherian commutative ring?

Answer 1

Your first ring $B$ is just $A[x^{\pm1}]$, the ring of Laurent polynomials with coefficients in $A$, which is the localization of $A[x]$ at the powers of $x$. It follows from this that $\dim B\leq\dim A[x]=\dim A+1$.

On the other hand, if $\def\p{\mathfrak p}\p$ is a prime in $A$ then $\mathfrak pB$ is a prime of $B$, easily seen to be the set of Laurent polynomials with coefficients in $\p$. If $\p_0\subsetneq\cdots\subsetneq\p_r$ is a chain of prime ideals in $A$, then $\p_0B\subsetneq\cdots\subsetneq\p_rB\subsetneq \p_rB+(x-1)B$ is a chain of prime ideals of $B$ one prime longer.

Conclusion: $\dim A[x^{\pm1}]=\dim A+1$.