Ring Product with Zero

Theorem
Let $\struct {R, +, \circ}$ 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 $\struct {R, +, \circ}$ is a ring, $\struct {R, +}$ is a group.

Since $0_R$ is the identity in $\struct {R, +}$, we have $0_R + 0_R = 0_R$.

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

Thus:

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