# Properties of ring epimorphisms that are true only over commutative rings

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.

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I believe that the property that you mention must be non-commutative, or why consider it as lef $R$-module? –  Murphy Mar 14 '13 at 1:00
I consider it as left module, because basically all my modules are left modules (unless I really need a right module). –  tj_ Mar 14 '13 at 6:10

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.

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Mark, thanks for the answer. Do you have a counter-example for the non-commutative case ? –  tj_ Mar 14 '13 at 6:12
@Moderators: Why has this answer (along with my question) been downvoted ? I've seen the discussion on downvotes on meta, and perhaps it's possible to find out some details about the downvoter by help of this new sample. –  tj_ Mar 14 '13 at 6:18
@TJ: Take a non-Hopfian finitely generated group (en.wikipedia.org/wiki/Hopfian_group) $G$, then its group ring $\mathbb{Z}G$ is not Hopfian and finitely generated. –  Mark Sapir Mar 14 '13 at 7:56
@TJ: I am not a moderator, but people downvote if they think they do not like a question or an answer. It is not surprising that among 10000 people reading this site there are a few people who do not like any particular question. It is actually quite surprising that not every question/answer is downvoted multiple times. –  Mark Sapir Mar 14 '13 at 8:14
Mark, this is true for any commutative Noetherian ring and does not require residual finiteness. See for instance my answer to mathoverflow.net/questions/71185/… –  Benjamin Steinberg Mar 15 '13 at 1:03

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.

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The example that Manny Reyes gave for a different purpose in the thread mathoverflow.net/questions/121468/… is also an example of this. –  Jeremy Rickard Mar 14 '13 at 11:44
Jeremy, thanks, that's exactly the example (Isbell: Epimorphisms and Dominions IV, J. London Math. Soc. (1969) s2-1 (1): 265-273. Cf. page 268, after 2.5). –  Ralph Mar 14 '13 at 19:55