nLab Krull dimension




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

P 1P 2P r P_1\subset P_2\subset\ldots \subset P_r

of distinct prime ideals in RR.

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

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 00
  • a poset has a deviation less than α\alpha for an ordinal α\alpha if for every descending chain
    a 1a 2a n a_1\geq a_2\geq \ldots\geq a_n\geq\ldots

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


Last revised on July 12, 2023 at 11:11:15. See the history of this page for a list of all contributions to it.