Does anyone know how to prove this:

Suppose that R is a ring. Show that -(a+b) = (-a)+(-b) for a, b in R.

Thanks guys. :)

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- February 21st 2006, 08:33 PMsuedenationAlgebra, Properties of a Ring
Does anyone know how to prove this:

Suppose that R is a ring. Show that -(a+b) = (-a)+(-b) for a, b in R.

Thanks guys. :) - February 22nd 2006, 04:07 AMtopsquarkQuote:

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).

-Dan - February 22nd 2006, 02:50 PMThePerfectHackerQuote:

Originally Posted by**topsquark**

- February 22nd 2006, 03:11 PMtopsquark
Pffl! That's three posts in a row I've got at least a detail wrong. I'm slipping! :eek:

Good catch, ThePerfectHacker.

-Dan