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. 



Properties of ring epimorphisms that are true only over commutative rings 
A commutative finitely generated ring is Hopfian (proved by Malcev) i.e. every surjective endomorphism is an automorphism. That is not true for non-commutative finitely generated rings. 


