# Proper Ideal iff Quotient Ring is Nontrivial

## 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