Main Index
Algebraic structures
Ring theory
Basic notions
Subject Index
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Main Index
Algebraic structures
Ring theory
Basic notions
Subject Index
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A proper ideal 
of a ring is called a prime ideal of 
if for any 
such that 
at least one of the factors 
and 
belongs to 
. The notion of the prime ideal generalizes the notion of a prime number. The principal ideal of 
generated by a prime number is prime.
Theorem. The factor ring 
of an ideal of a cumulative ring with identity is an integral domain if and only if the ideal 
is prime.
Theorem. In a commutative ring with identity, every maximal ideal is prime.
A prime ideal 
is called a minimal prime ideal over an ideal 
in a ring if there are no prime ideals of 
strictly contained in 
that contain 
. A prime ideal is called a minimal prime ideal of 
if it is a minimal prime ideal over the zero ideal 
.
The height of a prime ideal 
of a ring 
is the number of strict inclusions in the longest chain of prime ideals contained 
. Here, if
 | (1) |
is a chain of 
prime ideals contained in 
, where every inclusion is strict, then the height of 
is at least 
. We also say that the length of chain (1) is 
.
The height of an ideal 
is the infimum of the heights of all prime ideals containing 
.
Every principal ideal in a commutative Noetherian ring has height one.
If a ring 
has Krull dimension 
, then the polynomial ring 
has Krull dimension between 
and 
. If 
is a Noetherian ring, then the dimension of
is exactly 
.
In commutative algebra, the Krull dimension 1of a ring 
is defined as the number of strict inclusions in a maximal chain of prime ideals of 
. In other words, Krull dimension of a ring 
is the largest height of any prime ideal of 
, or it is the supremum of the lengths of chains of prime ideals
The set of all prime ideals of a commutative ring 
with identity is called prime spectrum of 
, and is denoted by 
, ↑
| 1 | Wolfgang Krull (1899 - 1971) a German mathematician, working in the field of commutative algebra  . |
Cite this web-page as:
Štefan Porubský: Prime ideal.