Definition 4

Let $\mathbf{A}$ be an integral domain. A prime ideal $\mathfrak{p}$ of $\mathbf{A}$ is said to be of height 1 if it is minimal among the non-zero prime ideals of $\mathbf{A}$.

We shall also say that the ideal $(0)$ is of height 0; a prime ideal of height $\leq 1$ is therefore by definition equal to $(0)$ or of height 1.

We shall define below, in a general way, the height of a prime ideal.