Chapter 5.1, Problem 14E

Let $R$ be a ring, and let $x,y,$ and $z$ be arbitrary elements of $R$. Complete the proof of Theorem $5.11$ by proving the following statements.

a. $x(-y)=-(xy)$

b. $(-x)(-y)=xy$

c. $x(y-z)=xy-xz$

d. $(x-y)z=xz-yz$

$$

Theorem $5.11$

$\square$

Additive Inverses and Products

For arbitrary $x,y,$ and $z$ in a ring $R$, the following equalities hold:

$(-x)y=-(xy)$

b. $x(-y)=-(xy)$

$(-x)(-y)=xy$

d. $x(y-z)=xy-xz$

$(x-y)z=xz-yz$