Proper Ideal iff Quotient Ring is Non-Null

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 $x - y \in A$:

$\mathfrak a \ne A$

$\blacksquare$

Also see

• Proper Ideal of Ring is Contained in Maximal Ideal