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? 

In a commutative ring, yes: let $u$ be such that $u(xy)=1 = (xy)u$. Then 
$$x(uy) = u(xy) = 1\quad\text{and}\quad (uy)x = u(xy) = 1,$$
so $x$ is invertible; and if $x$ is invertible, and $xy$ is a unit, then $y=x^{-1}xy$ is a product of units, hence a unit.
In a non-commutative ring, no; you can have a product be a unit yet neither factor be a unit. For example, let 
$$A = \prod_{i=1}^{\infty}\mathbb{Z}$$
and let $R$ be the ring of endomorphisms of $A$. Let $f\colon A\to A$ be the right-shift operator, and let $g\colon A\to A$ be the left-shift operator. Then $gf=1$ in $R$ and in particular the product is a unit, but neither $f$ nor $g$ are units ($g$ is not one-to-one, so it cannot be left-invertible, and $f$ is not onto, so it cannot be right-invertible).
Of course, if $xy$ is a unit in a non-commutative ring, then $x$ is right-invertible and $y$ is left-invertible, but that's the best you can say in general, as the example above shows.
