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 $$\left({R, +, \circ}\right)$$.