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?