give example of a submodule of a finitely generated module which is not finitely generated?

Question

you can take even only left R-modules

Answer 1

Take any non-Noetherian ring R (with 1) and consider a non-finitely generated ideal I of R. R is a finitely generated R-module (generated by 1), but I is not a finitely generated R-module. For a concrete example, take R=Z[x1,x2,x3,x4,...] where Z is the ring of integers and x1,x2,x3,... are independent transcendental elements (i.e. take the polynomial ring over the integers in infinitely many variables) and I=(x1,x2,x3,...).

Answer 2

R U Z

Answer 3

sorry dont know nothing about abstract algebra