you've been asking so many questions without showing us that you've thought about them for a minute! what makes me a little bit worried is that some of your questions are absolutely trivial

or just computational, which i've ignored, and some are fairly interesting! so, i really don't know if you basically have any idea about the questions?! i don't know about other members, who are

mostly nicer than me, but i'll keep helping you with those of your questions that are more interesting, however with less and less details! ok, regarding this question:

you should have mentioned that here is considered as an additive group, i.e. we want to prove that every finitely generated submodule of is cyclic. suppose is a finitely generated

submodule of so: for some let and: then: now we have: for somewhy?thus: