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In Advanced Modern Algebra, Rotman gives the following construction of a module that he denotes by $V^T$: Let $V$ be a vector space over a field $k$ and let $T:V \rightarrow V$ be a linear operator on $V$. Then, for a given polynomial $f(x) = \sum_i c_ix^i \in k[x]$ and $v \in V$ define scalar multiplication according to $$ \cdot :k[x] \times V \rightarrow V : f \cdot v \mapsto \sum_i c_i T^i(v) $$ where $T^i$ denotes the $i$-fold composition of $T$ with itself. It is clear that $V$ together with the indicated scalar multiplication is a $k[x]$-module.

My question is the following: Is this construction important enough to have a name and, if so, what is it called and how does it fit into the bigger picture? What is the motivation for this construction?

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    This is just an unpacking of the definition of a $k[x]$-module. Every $k[x]$-module looks like this.2012-05-01
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    Umm..ok. Well, I guess every $k[x]$ module would probably have to look something like this but that doesn't really get at the heart of my question nor does it address the role of $T$ which makes the construction a little more interesting. One can consider modules over any ring and since $k[x]$ is a ring it seems reasonable that one could consider modules over $k[x]$, but I'm trying to understand the motivation for this particular construction. Surely, there is something more than "this just follows from the definitions"2012-05-01
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    Every $k[x]$-module has to look _exactly_ like this, and it is an _unpacking of the definition_ that this is true. If this is not clear to you it is worth thinking about, and if you still have questions I can expand my comment into an answer.2012-05-01
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    More often than not the motivation of this module action is that we can say a lot of things about the linear mapping $T$ (or about all linear mappings) by using the fact that this is a module over $k[x]$. The polynomial ring being a PID gives us a number of useful tools.2012-05-01
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    One famous example of Jyrki's comment is that the Jordan decomposition of a linear operator is a corollary of the structure theorem of finitely generated modules over a PID.2012-05-01

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