* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

* Krull's theorem can fail for rings without unity.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.

The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).

In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.

# REDIRECT 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.

This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.

# REDIRECT Krull's principal ideal theorem

** Krull's intersection theorem

** Krull's principal ideal theorem

** Krull's theorem

* Krull's intersection theorem

* Krull's principal ideal theorem

Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.