Maximal ideal Assuming choice, commutative ring has a maximal ideal Let be a commutative ring and be a proper ideal. Then there exists a maximal ideal containing , invoking Zorn's lemma. #m/thm/ring ProofDefineas a Poset ordered by inclusion. Let be a chain, and define . Then for any , there exists a such that , and therefore , so is an ideal containing , and therefore an upper bound of in . By Zorn's lemma, has a maximal element. #state/tidy | #lang/en | #SemBr