# Quotient Ring Defined by Ring Itself is Null Ring

## Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\struct {R / R, +, \circ}$ be the quotient ring defined by $R$.

Then $\struct {R / R, +, \circ}$ is a null ring.

## Proof

From Ring is Ideal of Itself, it is clear we can form the quotient ring $\struct {R / R, +, \circ}$.

But $R / R = 0_R$ and so is the null ring.

$\blacksquare$