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 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.