additive inverse of the zero in a ring

In any ring $R$, the additive identity is unique and usually denoted by $0$. It is called the zero or neutral element of the ring and it satisfies the zero property under multiplication. The additive inverse of the zero must be zero itself. For suppose otherwise: that there is some nonzero $c\in R$ so that $0+c=0$. For any element $a\in R$ we have $a+0=a$ since $0$ is the additive identity. Now, because addition is associative we have

$0$$=$$a+0$

$=$$a+(0+c)$

$=$$(a+0)+c$

$=$$a+c.$

Since $a$ is any arbitrary element in the ring, this would imply that (nonzero) $c$ is an additive identity, contradicting the uniqueness of the additive identity. And so our suppostition that $0$ has a nonzero inverse cannot be true. So the additive inverse of the zero is zero itself. We can write this as $-0=0$, where the $-$ sign means “additive inverse”.

Yes, for sure, there are other ways to come to this result, and we encourage you to have a bit of fun describing your own reasons for why the additive inverse of the zero of the ring must be zero itself.

For example, since $0$ is the neutral element of the ring this means that $0+0=0$. From this it immediately follows that $-0=0$.