Let $A$ be the ring of $\mathbb{Z}^{\mathbb{N}}$ ring of integer-valued sequences (with addition and multiplication being the coordinate-wise operations), and let $N := \mathbb{Z}^{(\mathbb{N})}$ the subset of sequences with having only finitely many non-zero coefficients.
Show that $M := A$ is a finitely-generated $A$-module but $N$ is not a finitely-generated sub-$A$-module.
I’ve been stuck in the first part of this question for a bit, because I am unable to give a finite number of elements of $A$ that would be a generating set for A. Since A is a ring, the sub-A-modules are exactly the ideals. I am unable to show any stationary ascending chain conditions or generating elements of the ideals. I am guessing coordinate-reasoning might not work since we have an infinite number of coordinates. I would appreciate any hint.
For the second part, the generating elements would be elements of $N$ whose non-zero coordinate positions correspond to elements of $\mathscr{P} (\mathbb{N})$, which is uncountable, and hence not finitely-generated. I was wondering if there was any gap in reasoning or another way to argue this.
Given a ring $R$, a Module $M$ is finitely generated over $R$ if there are elements $m_{1},...,m_{n} $ such that every element of $m$ of $M$ can be written (not necessarly uniquely) as $\sum_{i=1}^{n} r_{i}m_{i}$.In particular, any ring $R$ is a finitely generated Module over itself , in fact it is a cyclic module (generated by 1,$R=R1$). Regarding your second question, what you are asked to prove is that no finite subset of $M$ can generate $M$,and not proving that $M$ can be gennerated by some infinite subset of it(otherwise any infinite Module wouldn't be finitely generated).So pick a finite subset $f_{1},...,f_{n} $ of elements of n,since each one of them has finite support,the for every $i\in \{1,2...,n\}$,there must exist $k_{i}$ such that $\forall p\geq k_{i},f_{i}(p)=0$,let $k$=max${k_{i}}$,then the (finitely supported) sequence f defined by : $f(n) =0 $ whenever $n\neq k+1$ and zero otherwise (which is an element of $N$,doesn't belong to the span of $f_{1},...,f_{n} $ .