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 Non-Null, $A / \mathfrak a$ is non-null.
By Krull's Theorem, $A / \mathfrak a$ has a maximal ideal.
By Correspondence Theorem for Quotient Rings, $A$ has a maximal ideal containing $\mathfrak a$.
$\blacksquare$