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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).

At present I am focussed on Chapter 2: Direct Sums and Short Exact Sequences.

Example 2.1.2 (i) on pages 38-39 reads as follows:https://www.physicsforums.com/attachments/2957

View attachment 2958In the above text B&K write:

" ... ... Clearly K is indecomposable as a module ... "

Can someone please explain exactly why this is the case? How can we demonstrate rigorously that K is indecomposable?

I have attempted to show that \(\displaystyle L \oplus N(x) = K^2\).

Can someone please critique my effort ... is it basically OK ...

Proof is as follows:

Let \(\displaystyle a \in L \oplus N(x)\)

Then \(\displaystyle a = l_1 + n_1 \)

\(\displaystyle = \left(\begin{array}{cc}1\\0\end{array}\right) k_1 + \left(\begin{array}{cc}x\\1\end{array}\right) k_2 \) where \(\displaystyle k_1, k_2 \in K\)

\(\displaystyle = \left(\begin{array}{cc}k_1\\0\end{array}\right) + \left(\begin{array}{cc}{k_2 x} \\k_2\end{array}\right)\)

\(\displaystyle = \left(\begin{array}{cc}{k_1 + k_2 x}\\k_2\end{array}\right) \in K^2 \)

Now let \(\displaystyle a \in K^2\); that is \(\displaystyle a = \left(\begin{array}{cc}{k_1}\\k_2\end{array}\right)\) for some \(\displaystyle k_1, k_2 \in K \)

Now take \(\displaystyle k_1 = c_1 + k_2 x \) where \(\displaystyle k_1, c_1, k_2\) and \(\displaystyle x \in K\). (This is permissible and possible since \(\displaystyle K\) is a field; \(\displaystyle c_1\), of course, may be negative)

Then \(\displaystyle a = \left(\begin{array}{cc}{c_1 + k_2 x }\\k_2\end{array}\right)\)

Therefore \(\displaystyle a = \left(\begin{array}{cc}{c_1 }\\0\end{array}\right) + \left(\begin{array}{cc}{ k_2 x }\\k_2\end{array}\right) \)

Therefore \(\displaystyle a = \left(\begin{array}{cc}{1 }\\0\end{array}\right) c_1 + \left(\begin{array}{cc}{ x }\\1\end{array}\right) k_2 \in L \oplus N(x) \)

Thus we conclude that \(\displaystyle L \oplus N(x) = K^2\).

Can someone please confirm that this is OK ... or alternatively amend/critique the argument ...

Hope someone can help.

Peter

B&K's definition of the internal direct sum is as follows:

View attachment 2959

View attachment 2960

B&K then point out that the definition of internal direct sum can be restated as follows:

https://www.physicsforums.com/attachments/2961

Finally, just before the example above, B&K define decomposable module, complement and indecomposable module as follows:

https://www.physicsforums.com/attachments/2962

At present I am focussed on Chapter 2: Direct Sums and Short Exact Sequences.

Example 2.1.2 (i) on pages 38-39 reads as follows:https://www.physicsforums.com/attachments/2957

View attachment 2958In the above text B&K write:

" ... ... Clearly K is indecomposable as a module ... "

**Question 1:**Can someone please explain exactly why this is the case? How can we demonstrate rigorously that K is indecomposable?

**Question 2:**I have attempted to show that \(\displaystyle L \oplus N(x) = K^2\).

Can someone please critique my effort ... is it basically OK ...

Proof is as follows:

Let \(\displaystyle a \in L \oplus N(x)\)

Then \(\displaystyle a = l_1 + n_1 \)

\(\displaystyle = \left(\begin{array}{cc}1\\0\end{array}\right) k_1 + \left(\begin{array}{cc}x\\1\end{array}\right) k_2 \) where \(\displaystyle k_1, k_2 \in K\)

\(\displaystyle = \left(\begin{array}{cc}k_1\\0\end{array}\right) + \left(\begin{array}{cc}{k_2 x} \\k_2\end{array}\right)\)

\(\displaystyle = \left(\begin{array}{cc}{k_1 + k_2 x}\\k_2\end{array}\right) \in K^2 \)

Now let \(\displaystyle a \in K^2\); that is \(\displaystyle a = \left(\begin{array}{cc}{k_1}\\k_2\end{array}\right)\) for some \(\displaystyle k_1, k_2 \in K \)

Now take \(\displaystyle k_1 = c_1 + k_2 x \) where \(\displaystyle k_1, c_1, k_2\) and \(\displaystyle x \in K\). (This is permissible and possible since \(\displaystyle K\) is a field; \(\displaystyle c_1\), of course, may be negative)

Then \(\displaystyle a = \left(\begin{array}{cc}{c_1 + k_2 x }\\k_2\end{array}\right)\)

Therefore \(\displaystyle a = \left(\begin{array}{cc}{c_1 }\\0\end{array}\right) + \left(\begin{array}{cc}{ k_2 x }\\k_2\end{array}\right) \)

Therefore \(\displaystyle a = \left(\begin{array}{cc}{1 }\\0\end{array}\right) c_1 + \left(\begin{array}{cc}{ x }\\1\end{array}\right) k_2 \in L \oplus N(x) \)

Thus we conclude that \(\displaystyle L \oplus N(x) = K^2\).

Can someone please confirm that this is OK ... or alternatively amend/critique the argument ...

Hope someone can help.

Peter

**Notes: B&K definitions and notation**

B&K's definition of the internal direct sum is as follows:

View attachment 2959

View attachment 2960

B&K then point out that the definition of internal direct sum can be restated as follows:

https://www.physicsforums.com/attachments/2961

Finally, just before the example above, B&K define decomposable module, complement and indecomposable module as follows:

https://www.physicsforums.com/attachments/2962

Last edited: