Please help with the following question given in the picture

Please help with the following question given in the picture below:

Recall that subtraction in a ring R is defined by x - y = x + (-y). Show that the following are true for any x, y, z epsilon R: x(-y) = (-x)y = -{xy) (-x)(-y) = xy x(y - z) = xy - xz -(x + y) = -x-y -(x-y) = -x + y

Solution

A commutative group with distributive propery with operators (+ & .) i.e. a.(b+c) = a.b + a.c is a ring.

Here x - y = x + (-y) is defined as substraction in a ring R.

(a) x*(-y) = x*(-1)*y = (-1)*x*y = -(xy) since xy = yx

again (-x)*y = (-1)*x*y = (-1)*(xy) =-(xy)

(b) (-x)*(-y) (-1)*x*(-1)*y = (-1)*(-1)*x*y due commutative property

= xy since (-1)*(-1) = 1

(c) x*(y-z) = xy - xz using the distributive property of a ring & subtraction

(d) - (x+y) = (-1)* (x+y) = (-1)*x + (-1)*y (using distributive property of ring)

= -x - y

(e) -(x-y) = (--1)* ( x+y) = (-1)*x + (-1)*(-1)*y ( Using distributive property of the ring

= -x -y since (-1)*(-1) = 1

=