Negatives, Differences, Reciprocals and Quotients

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $r$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, and let $x,y\in r$.

¶ The negative $-x$ is the unique $w\in r$ with $x+w=0$, given by Additive and Multiplicative Inverses Are Unique §negative, and the difference is $x-y=x+(-y)$.

¶ If $r$, with $+$, $\cdot$, $0$ and $1$, is a field and $y\neq0$, the reciprocal $y^{-1}$ is the unique $z\in r$ with $y\cdot z=1$, given by Additive and Multiplicative Inverses Are Unique §reciprocal, and the quotient is $x/y=x\cdot y^{-1}$.