Actually, it isn't unless the ring is commutative. In general the inverse of (a+b) is (-b)+(-a).Originally Posted bysuedenation

Calculate the following:

(a+b)+[(-b)+(-a)]=a+b+(-b)+(-a) by the associative property

=a+[b+(-b)]+(-a) again by associative

=a+0+(-a) by definition of inverse

=a+(-a) by the zero property

=0 by definition of inverse.

Thus (a+b)+[(-b)+(-a)] = 0.

Since the inverse of any element in the ring is unique, (-b)+(-a) = -(a+b).

-Dan