Krull Dimension

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals in that ring, a quantity that need not be finite even for a well behaved Noetherian ring. It was introduced to give an algebraic definition of the dimension of an algebraic variety, since the dimension of the affine variety defined by an ideal I in a polynomial ring R equals the Krull dimension of the quotient ring R modulo I. A field has Krull dimension zero, a polynomial ring in n variables over a field has Krull dimension n, and a principal ideal domain that is not itself a field has Krull dimension one.