16B70-AdditiveInverseOfTheZeroInARing.tex

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In any ring $R$, the additive identity is unique and usually denoted by $0$.  It is called the zero or {\em 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
\begin{eqnarray*}
0 & = & a + 0 \\ 
 & = & a + (0 + c) \\
 & = & (a + 0) + c \\
 & = & a + c.
\end{eqnarray*}
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 $-0 = 0$.
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