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? 

NO if $R$ is not commutative
Definition: Suppose $R$ is any ring with unity $1$. An element $u$ is said to be unit in $R$ if there exits $v\in R$ such that $uv=1$ and $vu=1$. 
Let $\mathbb{R}[x]$ be the (infinite dimensional) vector space of all polynomials over $\mathbb{R}$. Let $S$ denote the ring of all linear operators on $\mathbb{R}[x]$ with usual addition and composition of operators. 
Let $D\colon\mathbb{R}[x]\rightarrow\mathbb{R}[x]$ demote the differential operator: $D(p(x))=p'(x)$.
Let $J\colon:\mathbb{R}[x]\rightarrow \mathbb{R}[x]$ denote the integral operator: $J(a_0+a_1x+\cdots + a_nx^n)=a_0x+a_1\frac{x^2}{2}+\cdots + a_n\frac{x^{n+1}}{n+1}$.
Then $D\circ J$ is identity operator which is obviously unit. But $D$ is not unit: it is not one-to-one (why?) hence it can not have a two-sided inverse. 
