Quotient Ring Defined by Ring Itself is Null Ring

Theorem
Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$.

Let $\left({R / R, +, \circ}\right)$ be the quotient ring defined by $R$.

Then $\left({R / R, +, \circ}\right)$ is the null ideal of $\left({R, +, \circ}\right)$.

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

By Quotient Ring is an Ideal, $\left({R / R, +, \circ}\right)$ is an ideal of $R$.

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

Hence the result by definition of the null ideal of $\left({R, +, \circ}\right)$.