Elementwise Laws of a Commutative Ring

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

¶ $x+y\in R$, $x\cdot y\in R$, $0\in R$ and $1\in R$.

¶ $(x+y)+z=x+(y+z)$.

¶ $x+y=y+x$.

¶ $x+0=x$ and $0+x=x$.

¶ $(x\cdot y)\cdot z=x\cdot(y\cdot z)$.

¶ $x\cdot y=y\cdot x$.

¶ $x\cdot1=x$ and $1\cdot x=x$.

¶ $x\cdot(y+z)=x\cdot y+x\cdot z$ and $(x+y)\cdot z=x\cdot z+y\cdot z$.