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.
Famous quotes containing the words principal, ideal and/or theorem:
“Rather than have it the principal thing in my son’s mind, I would gladly have him think that the sun went round the earth, and that the stars were so many spangles set in the bright blue firmament.”
—Thomas Arnold (1795–1842)
“The tradition I cherish is the ideal this country was built upon, the concept of religious pluralism, of a plethora of opinions, of tolerance and not the jihad. Religious war, pooh. The war is between those who trust us to think and those who believe we must merely be led.”
—Anna Quindlen (b. 1952)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)