I've isolated a property of rings (integral domains, associative, unitary, non necessarily commutative) that is useful to me :


$$xR\cap yR\neq\{0\}\quad\text{ whenever $x$ and $y$ are non zero.}$$



> **Question.** Does this property has a name, or falls into a known and labelled class of rings ?


Commutative integral domains and division rings obviously have this property.



On rings $R$ such that $xR\cap yR$ is non zero whenever $x$ and $y$ are non zero
These rings are called [*right uniform rings.*](https://en.wikipedia.org/wiki/Uniform_module)


Generally, a right ideal $I$ is called (right) *uniform* if all nonzero right subideals $J,K\subseteq I$ have nonzero intersection: $J\cap K \neq 0$. 


Note: Your condition is equivalent to the condition that any two nonzero right ideals have nonzero intersection. 


