Proper Ideal iff Quotient Ring is Non-Null
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Theorem
Let $A$ be a commutative ring.
Let $\mathfrak a \subseteq A$ be an ideal.
The following statements are equivalent:
- $(1): \quad \mathfrak a$ is a proper ideal
- $(2): \quad$ The quotient ring $A / \mathfrak a$ is a non-null ring.
Proof
1 implies 2
Let $\mathfrak a$ be a proper ideal.
Then:
- $\exists x \in A \setminus \mathfrak a$
By definition of congruence modulo subgroup:
- $x + \mathfrak a \ne 0 + \mathfrak a$
in the quotient ring $A / \mathfrak a$.
Hence $A / \mathfrak a$ is a non-null ring.
$\Box$
2 implies 1
Let $A / \mathfrak a$ be a non-null ring.
Then:
- $\exists x, y \in A: x + \mathfrak a \ne y + \mathfrak a$
By definition of congruence modulo subgroup:
- $x - y \notin \mathfrak a$
Since $x - y \in A$:
- $\mathfrak a \ne A$
$\blacksquare$