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. In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... Wolfgang Krull (1899 - 1971) was a German mathematician, after whom Krull dimension, the Krull topology, and Krulls principal ideal theorem are named. ... In commutative algebra, the height of an ideal I in a ring R is the number of strict inclusions in the longest chain of prime ideals contained in I. In a Noetherian ring, Krulls height theorem says that the height of an ideal generated by n elements is no... 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: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...

Formally, if R is a Noetherian ring and I is a principal ideal of R, then I has height one.

This theorem can be generalized to ideals which are not principal, and the result is often called Krull's height theorem. It says, if R is a Noetherian ring and I is an ideal generated by n elements of R, then I has height at most n. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...