Krull's Theorem

Krull's theorem is a result in ring theory. It states every ring has a maximal ideal. It is equivalent to the axiom of choice. It is named for Wolfgang Krull, who stated the theorem first, in 1929.

Full statement and proof

Theorem. Let $R$ be a ring, and let $\mathfrak{a}$ be a (left, right, two-sided) ideal of $R$ that is distinct from $R$. Then there exists a maximal (left, right, two-sided) ideal of $R$ containing $\mathfrak{a}$.

Proof. Note that an ideal of $R$ is distinct from $R$ if and only if it does not contain 1. Let $\mathcal{F}$ be the family of proper ideals of $R$ containig $\mathfrak{a}$. Evidently, 1 is not an element of any member of this family, so the union of a totally ordered subset of this family is an element of the family. It then follows from Zorn's Lemma that $\mathcal{F}$ has a maximal element. $\blacksquare$

See also