Krull dimension

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In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. The Krull dimension need not be finite.

[edit] Explanation

If P₀, P₁, ... , P_n are prime ideals of the ring such that , then these prime ideals form a chain of length n. The Krull dimension is the supremum of the lengths of chains of prime ideals.

For example, in the ring (Z/8Z)[x,y,z] we can consider the chain

Each of these ideals is prime, so the Krull dimension of (Z/8Z)[x, y, z] is at least 3. In fact the dimension of this ring is exactly 3.

An alternate way of phrasing this definition is to say that the Krull dimension of R is the largest height of any prime ideal of R. (Dedekind domains that are not fields and discrete valuation rings have dimension one.) An integral domain is a field if and only if its Krull dimension is zero.

If a ring R has Krull dimension k, then the polynomial ring R[x] will have dimension at least k + 1 and at most 2k + 1. If R is Noetherian, then the dimension of R[x] is k + 1.

If K is a field and R is a finitely generated K-algebra, then R can be identified with the ring of polynomial functions on an affine variety X defined over K and the Krull dimension of R equals the usual dimension of the variety X.

There exists a ring with infinite Krull dimension even though every prime ideal has finite height.

[edit] See also

• Dimension theory (algebra)
• Regular local ring

[edit] References

• Irving Kaplansky, Commutative rings (revised ed.), University of Chicago Press, 1974, ISBN 0-226-42454-5. Page 32.
• A.I. Kostrikin and I.R. Shafarevich (edd), Algebra II, Encyclopaedia of Mathematical Scieinces 18, Springer-Verlag, 1991, ISBN 3-540-18177-6. Sect.4.7.