Definition 1

Let $\mathbf{A}, \mathbf{A}'$ be two rings and $h : \mathbf{A} \to \mathbf{A}'$ a ring homomorphism. An ideal $a' \in A'$ is said to lie above an ideal $a$ of $\mathbf{A}$ if $a = \overline{h}^1(a')$.

To say that a prime ideal $p'$ of $A'$ lies above an ideal $p$ of $\mathbf{A}$ therefore means that $p$ is the image of $p'$ under the continuous mapping "$h : \mathrm{Spec}(A') \to \mathrm{Spec}(A)$" associated with $h$ (Chapter 11, § 4, no. 3).

Note that for there to exist an ideal of $A'$ lying above the ideal $(0)$ of $\mathbf{A}$, it is necessary and sufficient that $h : \mathbf{A} \to \mathbf{A}'$ be injective.

Let $a$ be an ideal of $\mathbf{A}$; by taking quotients the homomorphism $h$ gives a homomorphism $h_1 : \mathbf{A}/a \to \mathbf{A}'/a'$; to say that $a'$ is an ideal of $A'$ lying above $a$ is equivalent to saying that $aA' \subset a'$ and that $a'/aA'$ is an ideal of $\mathbf{A}'/aA'$ lying above $(0)$.