If is a Ring (commutative with 1), the height of a Prime Ideal is defined as the Supremum of all so that there is a chain where all are distinct Prime Ideals. Then, the Krull dimension of is defined as the Supremum of all the heights of all its Prime Ideals.
See also Prime Ideal
References
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.