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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.