Does anyone know how to prove this:
Suppose that R is a ring. Show that -(a+b) = (-a)+(-b) for a, b in R.
Actually, it isn't unless the ring is commutative. In general the inverse of (a+b) is (-b)+(-a).Originally Posted by suedenation
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).