Krull's principal ideal theorem
Encyclopedia
In commutative algebra
, Krull's principal ideal theorem, named after Wolfgang Krull
(1899–1971), gives a bound on the height of a principal ideal
in a Noetherian ring
. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz
meaning "theorem").
Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.
This theorem can be generalized to ideal
s that are not principal, and the result is often called Krull's height theorem. This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then I has height at most n.
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, Krull's principal ideal theorem, named after Wolfgang Krull
Wolfgang Krull
Wolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in Baden-Baden, Imperial Germany and died in Bonn, West Germany.- See also :* Krull dimension* Krull topology...
(1899–1971), gives a bound on the height of a principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
in a Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz
Satz (disambiguation)
Satz is a German word and name, and may refer to:* Satz, a formal section in music analysis* "theorem" in German, conventionally used in the name of certain theorems. Note however that the word "Theorem" is also used in German, generally for more important results, and thus in a stricter...
meaning "theorem").
Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.
This theorem can be generalized to ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s that are not principal, and the result is often called Krull's height theorem. This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then I has height at most n.