Krull dimension • The Krull dimension of a commutative rin…

Krull dimension

• The Krull dimension of a commutative ring R is the supremum of the lengths of chains of prime ideals, a concept introduced by Wolfgang Krull.

• The Krull dimension can be infinite, even in Noetherian rings, and can also be applied to modules over non-commutative rings, defined as the deviation of the poset of submodules.

• It serves as an algebraic definition of the dimension of an algebraic variety, where the dimension of an affine variety defined by an ideal I in a polynomial ring R corresponds to the Krull dimension of R/I.

• A field has a Krull dimension of 0, while the polynomial ring k[x1, ..., xn] has a Krull dimension of n.

• A principal ideal domain that is not a field has a Krull dimension of 1, and a local ring has a Krull dimension of 0 if all elements of its maximal ideal are nilpotent.

• The length of a chain of prime ideals is determined by the number of strict inclusions, and the height of a prime ideal is the supremum of the lengths of chains of prime ideals contained within it.

• In Noetherian rings, every prime ideal has finite height, but some Noetherian rings can still have infinite Krull dimension.

• A ring is catenary if any inclusion of prime ideals can be extended to a maximal chain, and universally catenary if every finitely generated algebra over it is catenary.

• The height of an ideal is the infimum of the heights of all prime ideals containing it, which relates to the codimension of the corresponding subvariety in algebraic geometry.

• The Krull dimension of a ring is equivalent to the dimension of its spectrum as a topological space, representing the supremum of the lengths of chains of irreducible closed subsets.

• Examples include the ring of integers Z, which has a dimension of 1, and the zero ring, which is defined to have either dimension 0 or negative dimension.

• A Noetherian local ring is Cohen–Macaulay if its dimension equals its depth, with regular local rings serving as examples of such rings.

• The Krull dimension of a module over a commutative ring is defined in relation to the quotient of R that makes the module faithful, utilizing the annihilator.

• For non-commutative rings, the Krull dimension of a module is defined as the deviation of the poset of submodules ordered by inclusion, which may differ from the definition applicable to Noetherian rings.