Commutative Rings

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $r$ be a set, let $+$ and $\cdot$ be binary operations on $r$, written as in Binary Operations on a Set §notation with products formed before sums, and let $0,1\in r$.

¶ $r$, with $+$, $\cdot$, $0$ and $1$, is a commutative ring if, for all $x,y,z\in r$,

$$(x+y)+z=x+(y+z),\quad x+y=y+x,\quad x+0=x,\quad(x\cdot y)\cdot z=x\cdot(y\cdot z),\quad x\cdot y=y\cdot x,\quad x\cdot1=x,\quad x\cdot(y+z)=x\cdot y+x\cdot z,$$

and for every $x\in r$ there is $w\in r$ with $x+w=0$.