Ring theory MOC

Krull dimension

Let 𝑅 be a commutative ring and 𝔭 ◃𝑅 be a prime ideal. The height ht⁡𝔭 of 𝔭 is the supremum of the lengths of strictly increasing sequences of prime ideals culminating in 𝔭

𝔭0⊊𝔭1⊊⋯⊊𝔭ht⁡𝔭=𝔭

and the Krull dimension dim⁡𝑅 of 𝑅 is the supremum of the height of all prime ideals.

Properties

• Krull dimension of an integral domain

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