Krull's principal ideal theorem

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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.

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. It says, 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.