Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, and let $x,y,z\in R$.

¶ $0\cdot x=x\cdot0=0$.

¶ $-(-x)=x$, $-(x+y)=(-x)+(-y)$, $(-x)y=-(xy)$, $(-x)(-y)=xy$, $-(x-y)=y-x$ and $(x-y)z=xz-yz$.

¶ $(x+y)^{2}=x^{2}+2xy+y^{2}$, $(x-y)^{2}=x^{2}-2xy+y^{2}$ and $x^{2}-y^{2}=(x-y)(x+y)$.

¶ If $R$ is a field and $xy=0$, then $x=0$ or $y=0$.