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The minimum of the heights of the prime ideals containing the ideal. The height  $   \mathop{\rm ht} ( \mathfrak p ) $
of a prime ideal  $  \mathfrak p $
in a ring  $  A $
is the largest number  $  h $
(or  $  \infty $ if such a number does not exist) such that there exists a chain of different prime ideals

$$ 
\mathfrak p _ {0}  \subset   \mathfrak p _ {1}  \subset  \dots \subset  \ 
\mathfrak p _ {h}  =  \mathfrak p .
$$

The co-height  $   \mathop{\rm coht} ( \mathfrak p ) $
of a prime ideal  $  \mathfrak p $
is defined as the largest  $  h $
for which there exists a chain of prime ideals

$$ 
\mathfrak p  = \ 
\mathfrak p _ {0}  \subset   \mathfrak p _ {1}  \subset  \dots \subset   \mathfrak p _ {h}  \neq  A.
$$

Equivalently,

$$ 
 \mathop{\rm ht} ( \mathfrak p )  =   \mathop{\rm dim} ( A _ {\mathfrak p }  ),\ \ 
 \mathop{\rm coht} ( \mathfrak p )  =   \mathop{\rm dim} ( A / \mathfrak p ),
$$

where dim denotes the dimension of the corresponding Krull ring. The height of a prime ideal is equal to the codimension of the variety defined by the ideal, while the co-height equals the dimension of this variety. The height and the co-height of a prime ideal are connected by the inequality

$$ 
 \mathop{\rm ht} ( \mathfrak p ) +
 \mathop{\rm coht} ( \mathfrak p )
 \leq    \mathop{\rm dim}  A,
$$

which becomes an equality if, for example,  $  A $
is a local [[Cohen–Macaulay ring|Cohen–Macaulay ring]].

The prime ideals of height zero are the minimal prime ideals. The existence of prime ideals of height one in Noetherian integral domains is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. [[Krull ring|Krull ring]]). A more general result — Krull's theorem — interconnects the height with the number of generators of the ideal: In a Noetherian ring the height of an ideal generated by  $  r $
elements is not larger than  $  r $, 
and conversely: A prime ideal of height  $  r $
is the smallest of all prime ideals containing some  $  r $
elements. In particular, any ideal in a Noetherian ring has finite height; this is not true of the co-height [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,   "Primidealketten in allgemeinen Ringbereichen" , Berlin-Leipzig  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,   "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,   "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR></table>