# additive inverse of an inverse element

In any ring $R$, the additive inverse of an element $a\in R$ must exist, is unique and is denoted by $-a$. Since $-a$ is also in the ring $R$ it also has an additive inverse in $R$, which is $-(-a)$. Put $-(-a)=c\in R$. Then by definition of the additive inverse, $-a+c=0$ and $-a+a=0$. Since additive inverses are unique, it must be that $c=a$.

Title additive inverse of an inverse element AdditiveInverseOfAnInverseElement 2013-03-22 15:45:16 2013-03-22 15:45:16 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Result msc 16B70 InverseOfInverseInAGroup