Proper Ideal iff Quotient Ring is Non-Null

Theorem
Let $A$ be a commutative ring.

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


 * $(1): \quad \mathfrak a$ is a proper ideal
 * $(2): \quad$ The quotient ring $A / \mathfrak a$ is a non-null ring.

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.

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 $x - y \in A$:
 * $\mathfrak a \ne A$

Also see

 * Proper Ideal of Ring is Contained in Maximal Ideal