What is the name for the mathematical property involving addition or subtraction of fractions over a common denominator? For any number $x$ where $x\in\Bbb R$ and where $x\ne0$, what is the mathematical property which states that:
$${1-x^2\over x} = {1\over x} - {x^2\over x}$$

In $\Bbb R$ (or any other division ring), note that $\frac{x}{y} = x y^{-1}$, where $y^{-1}$ is defined as the number such that $yy^{-1}=1=y^{-1}y$, and $x-y = x + (-y)$, where $-y$ is defined as the number such that $y+(-y)=0=(-y)+y$, and that $(x+y)z = xz + yz$ (multiplication is left-distributive over addition)
So, in any context in which $b$ has an additive inverse, $c$ has a multiplicative inverse, and multiplication is left-distributive over addition, the expression becomes:
$$\frac{a-b}{c} = (a + (-b))c^{-1} = ac^{-1} + (-b)c^{-1} = \frac{a}{c} - \frac{b}{c}$$
Note that $0$ cannot have a multiplicative inverse, so this is true only when $c\ne0$
