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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 in order to fully understand Example 1.2.2 (iv) [page 16] ... indeed, I am somewhat overwhelmed by this construction ... ...
Example 1.2.2 (iv) reads as follows:View attachment 5089My question is as follows:
Why do Berrick and Keating bother to use the indeterminate $$T$$ in the above ... why not just use $$f(A)$$ ... ? what is the point of $$T$$ in the above example ...?
By the way ... I am assuming that $$f_0, f_1, \ ... \ ... \ f_r$$ are just elements of $$\mathcal{K}$$ ... ... is that correct?
Hope someone can help ...
Peter*** EDIT ***
It may make sense if we think of the polynomial $$f \in \mathcal{K} [T]$$ being evaluated at $$A$$ ... BUT ... when we evaluate a polynomial in $$\mathcal{K} [T]$$, don't we take values of $$T$$ in $$\mathcal{K}$$ ... ... but ... problem ... $$A$$ is an $$n \times n$$ matrix and hence (of course) $$A \notin \mathcal{K}$$ ... ?
... anyway, hope someone can explain exactly how the construction in this example "works" ...
Peter
I need help in order to fully understand Example 1.2.2 (iv) [page 16] ... indeed, I am somewhat overwhelmed by this construction ... ...
Example 1.2.2 (iv) reads as follows:View attachment 5089My question is as follows:
Why do Berrick and Keating bother to use the indeterminate $$T$$ in the above ... why not just use $$f(A)$$ ... ? what is the point of $$T$$ in the above example ...?
By the way ... I am assuming that $$f_0, f_1, \ ... \ ... \ f_r$$ are just elements of $$\mathcal{K}$$ ... ... is that correct?
Hope someone can help ...
Peter*** EDIT ***
It may make sense if we think of the polynomial $$f \in \mathcal{K} [T]$$ being evaluated at $$A$$ ... BUT ... when we evaluate a polynomial in $$\mathcal{K} [T]$$, don't we take values of $$T$$ in $$\mathcal{K}$$ ... ... but ... problem ... $$A$$ is an $$n \times n$$ matrix and hence (of course) $$A \notin \mathcal{K}$$ ... ?
... anyway, hope someone can explain exactly how the construction in this example "works" ...
Peter
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