# characterization of submodule of and of linear application.

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 $u'$.

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 ?

• No. I very much doubt the author of these questions intended you to regurgitate definitions as you are doing. In the first two questions, you can give a simple name to what they have described as submodules. Each one has a familiar and simple alternative name. The third one is not much different: you can describe the submodules because of the special structure of the ideals of $K[X]$ (oops, dropped a hint). I can't help with the 5th because I don't exactly understand what is written. – rschwieb Aug 1 '18 at 17:53
• @rschwieb If $V$ is a vector space over $K$, and $T \in L(V)$, then $V$ together with $T$ can be thought of as a $K[X]$-module where $X$ acts on a vector $v$ by $T$. That is, $Xv=T(v)$. The rest of the action is given from this. – Jonathan Dunay Aug 1 '18 at 19:51

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