Micheli Checkpoint 3.1.9. Let $R$ be a ring with unity. Using the ring axioms, show that $0_R\cdot x=0_R$ and that $-x=(-1)\cdot x.$

Proof. Since $R$ is an Abelian group, it has two additional properties we will need that follow from the ring axioms. The first property says that if $x+y=x+z$ then $y=z.$ Call this property "cancellation." The second property says that if $x+y=x$ then $y=0.$ Call this the "uniqueness" property, since it states that the ring's identity is unique.

To prove the first part of the checkpoint, apply the distributive property of $R$ to observe that $0x=(0+0)x=0x+0x,$ from which follows $0x=0$ by uniqueness. To prove the second part, apply the distributive property and the preceding result to note that $(-1)\cdot x + x$ $=(-1)\cdot x + (1\cdot x)$ $=(-1+1)x$ $=0x$$=0$ $-x+x.$ Since the preceding states that $-x+x=(-1)\cdot x + x,$ we have $-x=(-1)\cdot x$ by cancellation.