Dimension of ring

Last updated January 27, 2022

Definition

Definition. The dimension of a ring is the maximum length of an ascending chain of prime ideals $$ \mathfrak{p}_0\subsetneq\mathfrak{p}_1\subsetneq\cdots\subsetneq\mathfrak{p}_n\subsetneq A. $$

By the dimension of an ideal $I\subset A$, we mean the dimension of the quotient $A/I$.

Properties

1. If $A$ is an integral domain of dimension 0, then there are no prime maximal ideals except 0 and $A$.
2. (Krull ideal theorem) For $x\in A$, let $\mathfrak{p}$ be the minimal prime ideal containing $x$. Then $$\text{dim }A/\mathfrak{p}=\text{dim}(A)-1.$$