For the following modules I should give presentations by generators & relations (finitely many each if possible):
- The $\mathbb{Z}$-module $\mathbb{Q}$
- The $\mathbb{C}[X]$-module $\mathbb{C}^4$, where for the standard $\mathbb{C}$-basis $e_1,...,e_4$ of $\mathbb{C}^4$ we have
$$X*e_1 = 0, X*e_2 = e_1, X*e_3 = 0\ \text{and} \ X*e_4 = e_3$$
- The $\mathbb{C}[X_1,X_2,...]$-module $\mathbb{C}$, where $X_i*z = 0$ for all $z \in \mathbb{C}$ and for all $i \in \mathbb{N}_{>0}$
I see that $\mathbb{Q}$ cannot be generated by finitely many elements as an abelian group resp. $\mathbb{Z}$-module
I also think that $$X*e_1 = 0, \ X^2*e_2 = X*e_1 = 0, X*e_3 = 0\ \text{and} \ X^2*e_4 = X*e_3 = 0$$ can be seen as the relations I need in the second one. How do i write this down explicitly? As far as I know, a representation should be of the form of an exact sequence $$ R^{(J)} \xrightarrow{\phi_1} R^{(I)} \xrightarrow{\phi_2} M$$
However, I struggle with how to write it in this form.
Edit: What I've tried so far is the following for the first problem
Let $\mathbb{P}$ denote the set of prime numbers,
$\phi_1\colon \oplus_{p_i \in \mathbb{P}}\mathbb{Z}\to \oplus_{p_i \in \mathbb{P}}\mathbb{Z}$, $e_{p_i} \mapsto p_i*e_{p_i}-p_{i+1} * e_{p_{i+1}}$
$\phi_2\colon \oplus_{p_i \in \mathbb{P}}\mathbb{Z}\to \mathbb{Q}$, $e_{p_i} \mapsto 1/{p_i}$
The set of inverted primes generates $\mathbb{Q}$ and $\phi_2 \circ \phi_1 = 0$. Am I on the right path here?