I have two issues for which I seek solutions for a long time :
1- Submodules of finitely generated modules need not be finitely generated.
I searched and i found that the submodule $K$ of the finitely generated $R-module$ : $R = \mathbb{Z}[X1, X2, ...]$ consisting of all those polynomials with zero constant term is not finitely generated. The problem is (my problem is) that first I don't see how $R$ is equiped with the structure of a ring (commutative and unitary) and how is it finitely generated as an $R-module$. And secondly I don't understand the structure of $K$ and how is it not finitely generated.
2- Example of a non-free submodule and non-free quotient module of a free module.
And for that second one I don't even have a single clue.
Can anyone help me please. I desperately need some unswer.
Thank you in advance.