Proper Ideal of Ring is Contained in Maximal Ideal

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Theorem

Let $A$ be a commutative ring with unity.

Let $\mathfrak a \subseteq A$ be a proper ideal.


Then there exists a maximal ideal $\mathfrak m$ with $\mathfrak a \subseteq \mathfrak m$.


Proof

Let $A / \mathfrak a$ be the quotient ring.

By Proper Ideal iff Quotient Ring is Nontrivial, $A / \mathfrak a$ is non-trivial.

By Krull's Theorem, $A / \mathfrak a$ has a maximal ideal.

By Correspondence Theorem for Quotient Rings, $A$ has a maximal ideal containing $\mathfrak a$.

$\blacksquare$