Let $ A $ be the ring $ \mathbb C[X,Y]/(XY) $.
(a) Let $ S $ be an invertible $ d \times d $ matrix. Suppose we make $ V = \mathbb C^d $ and $ A $ module by having $ X $ act via the matrix $ S $. Find all possible matrices T via which $ Y $ can act in such a way that $ V $ becomes and $ A- $ module.
(b) Suppose $ S=\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$. Find all possible matrices $ T $ via which $ Y $ can act on $ V = \mathbb C^3 $ so that $ V $ becomes an $ A- $ module.
Not sure how to get started. Any help is much appreciated. Thanks in advance for any replies.
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(And thanks for taking the time to properly format and typeset your question right from the beginning).
I'll give a hint on (a), maybe that'll also help to solve (b): Suppose $V$ carries an $A$-module structure. Then, by the very definition, $X,Y\in A$ act on $V$ as ${\mathbb C}$-linear endomorphisms, say $\alpha,\beta$. Moreover, again following the definition, the relation $XY=0$ in $A$ between $X,Y$ must hold between their actions on $V$ as well, so $\alpha\circ\beta=0$ as endomorphisms of $V$. Now suppose that $\alpha$ is invertible on $V$; what does $\alpha\circ\beta=0$ then imply on $\beta$?
For (b), the strategy is the same, but the choice of possible $\beta$ is less restrictive. Note also that $\alpha$ and $\beta$ must commute, $\alpha\circ\beta = \beta\circ\alpha$, since the same holds in $A$.