TITLE: Properties of ring epimorphisms that are true only over commutative rings 
QUESTION [6 upvotes]: I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case.
Example: I learned from MO that if $R \hookrightarrow S$ is an epimorphism of commutative rings then $S/R$ is a torsion left $R$-module. But there are counter-examples to this property if $R$ is permitted to be non-comutative.

REPLY [4 votes]: For Artinian commutative rings $R$ all ring epimorphisms $R\to S$ are surjective. 
In general, this is false if $R$ is non-commutative. For, Isbell has constructed an epimorphism $R \to S$ where $A$ is finite (hence Artinian) and $S$ infinite. I'll have a look if I can find Isbell's example later on.