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I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).
I need help with Exercise 1.2.9 (a) ...
Exercise 1.2.9 (a) reads as follows:https://www.physicsforums.com/attachments/5101I am somewhat overwhelmed by this exercise ... can someone help me to get a start on this exercise ... ?
Hope someone can help ...
Some preliminary thoughts ... ...
$$A = \begin{pmatrix} k_{11} & ... & ... & ... & k_{1n} \\ . & & & & . \\ . & & & & . \\ . & & & & . \\ k_{n1} & ... & ... & ... & k_{nn} \end{pmatrix} $$ where $$k_{ij} \in \mathcal{K}$$
Now, $$M$$ is a $$\mathcal{K} [T]$$-module, so ... ...
... if $$m \in M$$ then the (right) action of the ring $$\mathcal{K} [T]$$ on the module $$M$$ is ...
$$m \cdot f(T) = m f_0 + Axf_1 + A^2 x f_2 + \ ... \ ... \ + A^r m f_r $$ ... ... ... (1)
where
$$f(T) = f_0 + f_1 T + f_2T^2 + \ ... \ ... \ + f_r T^r$$ ... ... ... (2)
(Note: despite some help on this issue I still do not completely understand how the $$A^n$$ end up on the left in (1) above ... can someone please help?)Now ... proceeding ... ... $$L$$ is a submodule of $$M$$ ... BUT ... what does it mean that $$L$$ has a subspace $$U$$ ... what is being said here ...?
Can someone please help me to proceed ... ?
Help will be much appreciated ... ...
Peter
I need help with Exercise 1.2.9 (a) ...
Exercise 1.2.9 (a) reads as follows:https://www.physicsforums.com/attachments/5101I am somewhat overwhelmed by this exercise ... can someone help me to get a start on this exercise ... ?
Hope someone can help ...
Some preliminary thoughts ... ...
$$A = \begin{pmatrix} k_{11} & ... & ... & ... & k_{1n} \\ . & & & & . \\ . & & & & . \\ . & & & & . \\ k_{n1} & ... & ... & ... & k_{nn} \end{pmatrix} $$ where $$k_{ij} \in \mathcal{K}$$
Now, $$M$$ is a $$\mathcal{K} [T]$$-module, so ... ...
... if $$m \in M$$ then the (right) action of the ring $$\mathcal{K} [T]$$ on the module $$M$$ is ...
$$m \cdot f(T) = m f_0 + Axf_1 + A^2 x f_2 + \ ... \ ... \ + A^r m f_r $$ ... ... ... (1)
where
$$f(T) = f_0 + f_1 T + f_2T^2 + \ ... \ ... \ + f_r T^r$$ ... ... ... (2)
(Note: despite some help on this issue I still do not completely understand how the $$A^n$$ end up on the left in (1) above ... can someone please help?)Now ... proceeding ... ... $$L$$ is a submodule of $$M$$ ... BUT ... what does it mean that $$L$$ has a subspace $$U$$ ... what is being said here ...?
Can someone please help me to proceed ... ?
Help will be much appreciated ... ...
Peter
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