Corollary

Let $\rho : A \to B$ be an injective homomorphism of finitely generated k-algebras. One has $\dim(A) \leq \dim(B)$.

Indeed, let $p$ be a minimal prime ideal of $A$ such that $\dim(A) = \dim(A/p)$. There exists a prime ideal $q$ of $B$ above $p$ (II, § 2, no 6, prop. 16). By prop. 4 applied to $A/p$ and $B/q$, one has $\dim(A) = \dim(A/p) \leq \dim(B/q) \leq \dim(B)$, whence the corollary.