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How do you see this? $End_{\mathbb{C}} ( \mathbb{C}[x])$

As $M_{n}(\mathbb{C})=End_{\mathbb{C}} ( \mathbb{C}[x])$, so it just a matric with basis of polyonial?

Take the weyl algebra $A_1=\{ {\sum_{i=0}^{n} f_i}(x) \partial^i \ : f_{i}(x) \in \mathbb{C}, n\in \mathbb{N}\}$.

It says this weyl algebra A_1 is a subring of $End_{\mathbb{C}} ( \mathbb{C}[x])$so does that mean weyl algebra just a big matrix with complex numbers in them?

Also, is this the hardest example of a non commutative ring? As I hate to see tricker stuff than this :<

I used to like rings.

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    Fixed it. Was being silly.2011-10-06

1 Answers 1

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I am sort of guessing what the question actually is, but here is the answer to my guessed question:

$\text{End}_\mathbb{C}(\mathbb{C}[x])$ is just the space of $\mathbb{C}$-linear maps on the polynomial ring. To verify that a map is $\mathbb{C}$-linear is usually trivial. Yes, you can think of an endomorphism as an infinite dimensional matrix (not $n$-dimensional!), but I don't think that that's the easiest way to think about them.

Now, the Weyl algebra is defined as $A_1=\{ {\sum_{i=0}^{n} f_i}(x) \partial^i \ : f_{i}(x) \in \mathbb{C}[x]\}$ (and not $f_i\in \mathbb{C}$), where $\partial^i$ is the $i$-th derivative operator. Clearly, multiplication by a polynomial is linear, and clearly differentiation is $\mathbb{C}$-linear. Compositions and sums of linear maps are linear, so the elements of the Weyl algebra are indeed $\mathbb{C}$-endomorphisms of $\mathbb{C}[x]$. That's all there is to it.

Finally, no, that's not the "hardest example" (whatever that may mean) of a non-commutative ring.

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    Oh I see. That makes a lot more sense now.2011-10-06