Krull's theorem can fail for rings without unity.

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

See Krull's theorem at maximal ideal.

Another theorem commonly referred to as Krull's theorem: Let be a Noetherian ring and an element of which is neither a zero divisor nor a unit.

Thus, by a variant of Krull's theorem, there exists a prime ideal P that contains I and is still disjoint from S. (see prime ideal.)

Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal) which is a direct consequence of Krull's theorem.

In mathematics, more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, proves the existence of maximal ideals in any commutative unital ring.

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)[X] is therefore an integral domain.