characterization of submodule of and of linear application.

Question

Let $A$ a commutative ring, $G$ an abelian group and $K$ a field.

1) Characterize submodules of $A$ for its natural structure of $A$-module

2) Characterize submodules of $G$ for its natural structure of $\mathbb Z$-module

3) Characterize submodules of $K[X]$ for its natural structure of $K[X]$-module.

4) Let $G'$ an abelian group. Characterize linear applications $G\to G'$ for their structure of $\mathbb Z$-module.

5) Let $E$ and $E'$ two $K$-vectors spaces, $u\in \mathcal L(E)$ and $u'\in \mathcal L(E')$. Characterize linear application $E\to E'$ for the structure of $K[X]$-module associated to $u$ and $u'$.

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I'm not really sure what to do.

1) $S$ is a submodule of $A$ if $0\in S$, $as\in S$ for all $a\in A$ and all $s\in S$ and $s+s'\in S$ for all $s,s'\in S$ ? Is it the question ?

2) $H$ is a submodule if $0\in H$, $kg\in H$ for all $k\in \mathbb Z$ and all $g\in G$ and $h+h'\in H$ for all $h,h'\in H$ ? Is it the question ?

3) $S[X]$ is a $K[X]$-submodule if $0\in S[X]$, $p(x)+q(x)\in S[X]$ for all $p(x),q(x)\in S[X]$ and $p(x)q(x)\in S[X]$ for all $p(X)\in K[X]$ and all $q(x)\in S[X]$ ? Is it the question ?

4) A linear map $f:G\to G'$ is a map s.t. $f(kg)=kf(g)$ for all $k\in \mathbb Z$ and all $g\in G$ and $f(g+h)=f(g)+f(h)$ for all $g,h\in G$ ? Is it the question ?

5) $f:E\to E'$ is linear if $f(p(x) y)=p(x)f(y)$ for all $p(x)\in K[X]$ and all $y\in E$ and $f(y+y')=f(y)+f(y')$ for all $y,y'\in E$ ? Is it the question ?

Answer 1

It appears that the questions are asking you to discuss these in already familiar terms. For example, for 2), the answer would be that the submodules of $G$ considered as a $\Bbb{Z}$-module are just the subgroups of $G$.

For the last one, I don't think it is that straightforward. I think they want you to write down a condition on the map $f$ using $u$ and $u'$ that is necessary and sufficient to guarantee that the map is $K[X]$-linear (I know that is vague).

Hint:

If $f$ is a $K$-linear map and for each $y \in E$, $f(Xy)=Xf(y)$, then $f$ is $K[X]$-linear.

The condition I have in mind is that:

$f$ is a $K$-linear and $f\circ u = u' \circ f$