If $V$ is a $\mathbb{C}$-vector space, and $a\in End \ V $, then we let $V_a$ be the $\mathbb{C}[t]$-module with ground set $V$ and scaling defined by: $Pv=P(a)v, \ \ (P\in \mathbb{C}[t], \ v\in V)$
i) Suppose $M$ is a $\mathbb{C}[t]$-module. Then there exist $V,a$ and a $\mathbb{C}[t]$-module isomorphism $f:M\rightarrow V_{a}$.
ii) Suppose $R= \mathbb{C}[t]/t^{2}\mathbb{C}[t]$ and let $M$ be a $R$-module. Then there exist $V,a$ so that $a^{2}=0$ and also a $R$-module isomorphism $f:M\rightarrow V_{a}$.
i) To show that this is true, we must find a module homomorphism which is one-to-one and onto. If $P(t)= b_{n}t^{n}+b_{n-1}t^{n-1}+...b_{1}t+b_{0}$ and one puts: $P(a)=b_{n}a^{n}+b_{n-1}a^{n-1}+...+b_{1}a+b_{0}I$, then $a^{n}$, where I is the identity and $a^{n}$ is the composition of a with itself ($va^{n}=(...(va)a...)a)$ This is then made into a $\mathbb{C}[t]$ module by: $P(t)v = P(a)v$.
How to construct the isomorphism $f: M\rightarrow V_{a}$?
ii) I know that there must be a vector space with $a^{2}=0$ because the factorial ring contains $t^{2}\mathbb{C}[t]$, but I don't see how to show that it exists and also not how to construct a module isomorphism?
Would be glad if somebody could answer my questions.