Every proper ideal is contained in a maximal ideal

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Statement

Let R be a commutative unital ring. Then, every proper ideal (i.e. every ideal that is not the whole ring) is contained in a maximal ideal.

Explanation

The corresponding statement is not true in a number of other algebraic structure, primarily because the condition of being proper cannot be tested through the presence or absence of a single element. For instance, it is not in general true for modules over a fixed commutative ring, and it is not true in general for groups. The key difference in rings is that the presence or absence of a single element can be used to test for properness.

Proof

The proof uses the axiom of choice. We present here three slightly different formulations: transfinite induction, Zorn's lemma and the Hausdorff maximality principle.

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