What about the Homomorphisms in Problem 4? Namely,
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Is it true that:
\[ \operatorname {Hom}_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=k\cdot \operatorname {id} \]
is a one-dimensional vector space, or are there more \(k[x, y]\)-module
homomorphisms than just the multiples of the identity?
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Are there any non-split extensions of \(k[x, y]\)-modules:
\[ 0\to \langle x,y\rangle \to M\to \langle x,y\rangle \to 0 \]
i.e. is
\(\operatorname {Ext}^1_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=0\) or not?