nLab Krull dimension

Definition

The Krull dimension of a commutative ring $R$ is the supremum of lengths of chains

$$P_1\subset P_2\subset\ldots \subset P_r$$

of distinct prime ideals in $R$.

If $R$ is a possibly noncommutative ring $R$ and $M$ a left $R$-module, then the Krull dimension of $M$ is by definition a deviation of the poset of subobjects of $M$.

A deviation of a poset is defined recursively.

• the deviation of a trivial poset is $-\infty$
• the deviation of a poset satisfying a descending chain condition is $0$
• a poset has a deviation less than $\alpha$ for an ordinal $\alpha$ if for every descending chain
  $$a_1\geq a_2\geq \ldots\geq a_n\geq\ldots$$

  the subposets of all elements between $a_n$ and $a_{n+1}$ do not have deviation of less than $\alpha$ for at most finitely many $n$.

Literature

• wikipedia: deviation of a poset