View source for Height of an ideal

The minimum of the heights of the prime ideals containing the ideal. The height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468301.png" /> of a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468302.png" /> in a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468303.png" /> is the largest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468304.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468305.png" /> if such a number does not exist) such that there exists a chain of different prime ideals

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468306.png" /></td> </tr></table>

The co-height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468307.png" /> of a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468308.png" /> is defined as the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468309.png" /> for which there exists a chain of prime ideals

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683010.png" /></td> </tr></table>

In other words,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683011.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683012.png" /></td> </tr></table>

which becomes an equality if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683014.png" /> elements is not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683015.png" />, and conversely: A prime ideal of height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683016.png" /> is the smallest of all prime ideals containing some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683017.png" /> 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>