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:
“It is perhaps the principal admirableness of the Gothic schools of architecture, that they receive the results of the labour of inferior minds; and out of fragments full of imperfection ... raise up a stately and unaccusable whole.”
—John Ruskin (1819–1900)
“Our ideal ... must be a language as clear as glass—the person looking out of the window knows there is glass there, but he is not concerned with it; what concerns him is what comes through from the other side.”
—Elizabeth Bowen (1899–1973)
“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)