Krull's principal ideal theorem

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 ideals 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.

References

• Matsumura, Hideyuki (1970), Commutative algebra, New York: Benjamin , see in particular section (12.I), p. 77