If $xy$ is a unit, are $x$ and $y$ units? I know if $x$ and $y$ are units, in say a commutative ring, then $xy$ is a unit with $(xy)^{-1}=y^{-1}x^{-1}$. 
But if $xy$ is a unit, does it necessarily follow that $x$ and $y$ are units? 

Yes. Let $z=xy$. If $z$ is a unit with inverse $z^{-1}$, then $x$ is a unit with inverse $yz^{-1}$, and $y$ is a unit with inverse $xz^{-1}$, because
$$x(yz^{-1})=(xy)z^{-1}=zz^{-1}=1$$
$$y(xz^{-1})=(yx)z^{-1}=(xy)z^{-1}=zz^{-1}=1$$
