Ring Product with Zero

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

Then:


 * $$\forall x \in R: 0_R \circ x = 0_R = x \circ 0_R$$

That is, the zero is a zero element for the ring product, thereby justifying its name.

Proof
Because $$\left({R, +, \circ}\right)$$ is a ring, $$\left({R, +}\right)$$ is a group.

Since $$0_R$$ is the identity in $$\left({R, +}\right)$$, we have $$0_R + 0_R = 0_R$$.

From the Cancellation Laws, all group elements are cancellable, so every element of $$\left({R, +}\right)$$ is cancellable for $$+$$.

Thus:

$$ $$ $$ $$

Next:

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