Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can *fail* to be abelian categories. 


Poking around on the internet has lead me to believe that these categories will agree and be abelian if R is a (left?) Noetherian ring, but I'm interested in the more general case. I've found warnings that these categories can fail to be abelian but haven't found much discussion of what goes wrong. I feel like there should be some really good illuminating examples.


I think it's not too hard to show that the cokernel of a map of finitely presented modules is again finitely presented. So one of the things I'm really looking for is an example of a map of between finitely presented modules in which the kernel is *not* finitely presented. It would be even better if the kernel was not finitely generated. Is this possible?



Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories? 
[Background to the following can be found in Lam's *Lectures on Modules and Rings*, section 4.G.]


**Definition:** A finitely generated (f.g.) right module $M\_R$ is *coherent* if every f.g. submodule of $M$ is finitely presented (f.p.). 


Now let $M\_R$ be a finitely presented module that is not coherent. There exists some submodule $N\_R\subseteq M$ that is f.g. but not f.p. Pick a surjection $R^n\twoheadrightarrow N$, and consider the composition $f\colon R^n \twoheadrightarrow N\hookrightarrow M$. Because $N\cong R^n/\ker(f)$ is not f.p., it must be the case that $\ker(f)$ is not f.g. 


So all we need is an explicit example of a f.p. module that is not coherent. For this, see Jack Schmidt's answer to this question. Using my approach in his case, instead of looking at a surjection $R\twoheadrightarrow R/I$, we would look at the composition $R^n\twoheadrightarrow I\hookrightarrow R$.


