# Finitely generated modules over a PID, and normed polynomials

I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve:

14.$\text{}$ i) If $R$ is a PID, $M=Rx_{1}+...+Rx_{N}$ a finitely generated $R$ module, $L\subset M$ a submodule of $R$. Then $L=Ry_{1}+...+Ry_{J}$, where $J\le N$ is also finitely generated.

ii) If $\mathbf{F}$ is a finite field with $q$ elements, and $N_{d} = N_{d}(\mathbf{F})$ is the number of normed polynomials $P \in \mathbf{F}[t]$ with degree $d$. What are $N_{2}$, $N_{3}$ ?

If there is anybody who understands these exercises, I would be very glad if you could explain them to me. Because when my professor tried it, I don't seem to have understood. Thank you for your attention.

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Dear Arthur: Welcome to MSE! I think you should ask your two questions separately. I also suggest that you change your user name to something like "Arthur from Belgium". – Pierre-Yves Gaillard Dec 22 '11 at 14:22

If $M$ is an $R$-module generated by $n$ elements, and if $L$ is a sub-module of $M$, then $L$ can also be generated by $n$ elements.
Clearly, the rank of $L$ is at most that of $M$.
Thus we can assume that $M$ is torsion, and even that $M$ is annihilated by some power of a prime element $p$.
Then the statement follows in this case from the fact that the minimum number of generators of $M$ is the dimension of the $R/(p)$-vector space formed by the elements $x$ of $M$ satisfying $px=0$.