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Krull Dimension

If $R$ is a Ring (commutative with 1), the height of a Prime Ideal $p$ is defined as the Supremum of all $n$ so that there is a chain $p_0\subset\cdots p_{n-1}\subset p_n=p$ where all $p_i$ are distinct Prime Ideals. Then, the Krull dimension of $R$ is defined as the Supremum of all the heights of all its Prime Ideals.

See also Prime Ideal


Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry. New York: Springer-Verlag, 1995.

Macdonald, I. G. and Atiyah, M. F. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

© 1996-9 Eric W. Weisstein