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I'm currently trying to figure out why $\mathbb{C}[x,y] \ncong (x,y) \oplus \mathbb{C}[x,y]/(x,y)$ where $\mathbb{C}[x,y]$ acts on itself by a strange action which I describe below (this action forces the torsion part of the module $\mathbb{C}[x,y]$ to be $\mathbb{C}$). From what I understand, $\mathbb{C}[x,y] \cong (x,y) + \mathbb{C}[x,y]/(x,y)$, however I am struggling to see why this sum is not direct.

The action is:

$a*f(x,y)=af(x,y), \ a \in \mathbb{C},$

$x*f(x,y) = (\partial_yf)(0,0)+x(f(x,y)-f(0,0)),$

$y*f(x,y) = (-\partial_xf)(0,0)+y(f(x,y)-f(0,0))$.

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    How do you even define $A+M/A$? (I suppose you mean $A+(M/A)$, not $(A+M)/A$?) – 2017-02-12
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    Yes, that is correct. Sorry for the confusion, I will edit the title. – 2017-02-12

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