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:
“I note what you say of the late disturbances in your College. These dissensions are a great affliction on the American schools, and a principal impediment to education in this country.”
—Thomas Jefferson (1743–1826)
“And into the gulf between cantankerous reality and the male ideal of shaping your world, sail the innocent children. They are right there in front of us—wild, irresponsible symbols of everything else we can’t control.”
—Hugh O’Neill (20th century)
“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)