Proper Ideal iff Quotient Ring is Nontrivial
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This theorem requires a proof..
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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
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