In a ring, how do we prove that a * 0 = 0? In a ring, I was trying to prove that for all $a$, $a0 = 0$.
But I found that this depended on a lemma, that is, for all $a$ and $b$, $a(-b) = -ab = (-a)b$.
I am wondering how to prove these directly from the definition of a ring.
Many thanks!

Proceed like this


*

*$a0 = a(0+0)$, property of $0$.

*$a0 = a0 + a0$, property of distributivity.

*Thus $a0+ (-a0) = (a0 + a0) +(-a0)$, using existence of additive inverse.

*$a0+ (-a0) = a0 + (a0 + (-a0))$ by associativity.

*$0 = a0 + 0$ by properties of additive inverse.

*Finally $0 = a0$ by property of $0$.


Your lemma is also true, you can now prove it easily: 
Just note that $ab +a(-b)= a(b + (-b))= a0= 0$.
