Proper Ideal iff Quotient Ring is Nontrivial

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Theorem

Let $A$ be a commutative ring.

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


The following are equivalent:

$(1): \quad \mathfrak a$ is a proper ideal
$(2): \quad$ The quotient ring $A / \mathfrak a$ is nontrivial ring


Proof