Krull's principal ideal theorem

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".

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".

References


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