Free Modules - Bland - Proposition 2.2.3

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.3.

Proposition 2.2.3 and its proof read as follows:View attachment 3529In the text above Bland writes:

" ... ... The proof of the equivalence of (1) and (3) is equally straightforward and so is omitted. ... ... "Despite Bland's assurance that the proof is straightforward, I am unsure of how to frame a rigorous and formal proof of the equivalence of (1) and (3) ... can someone please help me in this matter ...I assume the relevant direct sum in (3) is the external direct sum and so I am providing the relevant definition from Bland's text, as follows:View attachment 3530

I am assuming that the proof will have to draw on the isomorphism between sums of the form \(\displaystyle \sum_{\Delta} x_\alpha a_\alpha\) and tuples such as \(\displaystyle ( a_\alpha )\) ... just thinking anyway ...

Hope someone can help ... ...

Peter

 

[caffeinemachine]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.3.

Proposition 2.2.3 and its proof read as follows:View attachment 3529In the text above Bland writes:

" ... ... The proof of the equivalence of (1) and (3) is equally straightforward and so is omitted. ... ... "Despite Bland's assurance that the proof is straightforward, I am unsure of how to frame a rigorous and formal proof of the equivalence of (1) and (3) ... can someone please help me in this matter ...I assume the relevant direct sum in (3) is the external direct sum and so I am providing the relevant definition from Bland's text, as follows:View attachment 3530

I am assuming that the proof will have to draw on the isomorphism between sums of the form \(\displaystyle \sum_{\Delta} x_\alpha a_\alpha\) and tuples such as \(\displaystyle ( a_\alpha )\) ... just thinking anyway ...

Hope someone can help ... ...

Peter

It seems that the notation is throwing you off.

Here's something you should try.

Take $\Delta$ finite and $R$ a field.

So we are just talking about vector spaces.

Can you do it in this very special case?

Now do it for the case when $R$ is an arbitrary ring.

Finally take $\Delta$ to be an arbitrary index set.

Tell me if you still get stuck.

 

[Math Amateur]

It seems that the notation is throwing you off.

Here's something you should try.

Take $\Delta$ finite and $R$ a field.

So we are just talking about vector spaces.

Can you do it in this very special case?

Now do it for the case when $R$ is an arbitrary ring.

Finally take $\Delta$ to be an arbitrary index set.

Tell me if you still get stuck.

Thanks caffeinemachine ... will give your idea a try ...

Just by by the way, I have a suspicion that you are correct ... that part of my problem is with getting a good feel for and understanding of the notation ...

Peter

 

[Math Amateur]

It seems that the notation is throwing you off.

Here's something you should try.

Take $\Delta$ finite and $R$ a field.

So we are just talking about vector spaces.

Can you do it in this very special case?

Now do it for the case when $R$ is an arbitrary ring.

Finally take $\Delta$ to be an arbitrary index set.

Tell me if you still get stuck.

Hi caffeinemachine,

I think you were right, as indicated above, in pointing to difficulties in notation ... indeed, I need a preliminary clarification of the meaning of \(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\) before starting on a proof of the statement that

\(\displaystyle (1) \ \ \{ x_\alpha \}_\Delta\) is a basis for \(\displaystyle M \ \Longleftrightarrow \ (3) \ \ M = \bigoplus_{\Delta} x_\alpha R\)

with a finite number of elements in \(\displaystyle \Delta\) ... ... (your suggestion of trying with a finite number of elements in \(\displaystyle \Delta\)!)Now ... ... (problem 1!) I am not sure whether Bland means an external or an internal direct sum by \(\displaystyle \bigoplus_{\Delta} x_\alpha R\) ... ... but maybe it does not matter because the internal and external direct sums are isomorphic ... ? (what do you think?)Bland gives the definition of an external direct sum as follows:View attachment 3555Now, we are dealing with a case where

\(\displaystyle M = \bigoplus_{\Delta} x_\alpha R \)

\(\displaystyle = \{ ( x_\alpha r_\alpha ) \in \prod_\Delta x_\alpha R \ | \ x_\alpha = 0 \text{ for almost all } \alpha \in \Delta \}\)

BUT ... we are taking the finite case

\(\displaystyle \Delta = \{ x_1, x_2, x_3 \} \)

so we can write

\(\displaystyle M = \bigoplus_{\Delta} N_\alpha\)

\(\displaystyle = \{ ( x_\alpha r_\alpha ) \in N_1 \times N_2 \times N_3 \}\)

\(\displaystyle = \{ (x_i r_i ) \ | \ x_i r_i \in N_i \text{ where } i = 1,2,3 \} \)

\(\displaystyle = \{ (x_1 r_1, x_2 r_2, x_3 r_3) \ | \ r_1, r_2, r_3 \in R \} \)
Now the above seems OK to me ... ... BUT... ... in another post, Euge has indicated that this is incorrect ... ...so it very probably IS incorrect! ... ... but I am not sure why ... ... however, if you can see why it is incorrect, then please let me know so I can proceed with the proof of \(\displaystyle (1) \ \Longleftrightarrow \ (3)\) without delay ... ...
Now ... ... suppose that Bland means the internal direct sum by the term \(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\) ... ... then we have the following definition in Bland ... ...View attachment 3556

So then, assuming that we have the condition ( ? do we ? )

\(\displaystyle M_\beta \ \bigcap \ \sum_{ \alpha \ne \beta } M_\alpha = 0 \text{ for each } \beta \in \Delta\) , and taking a finite set \(\displaystyle \Delta = \{ x_1, x_2, x_3 \} \),

and following the definition,

we have:

\(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\)

\(\displaystyle = x_1R + x_2R + x_3R
\)

\(\displaystyle = N_1 + N_2 + N_3\), say

\(\displaystyle = \sum_\Delta x_i r_i\) since the elements of \(\displaystyle N_i\) are of the form \(\displaystyle x_i r_i\)
Now ... which should I assume for the proof of \(\displaystyle (1) \ \Longleftrightarrow \ (3)\) ... ... the external direct sum or the internal direct sum ... ... can you help?Can you also please critique my analysis above? Does it make sense? Are there any erroneous statements?

Hope you can help ... ...

Peter

 

[caffeinemachine]

Hi caffeinemachine,I think you were right, as indicated above, in pointing to difficulties in notation ... indeed, I need a preliminary clarification of the meaning of \(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\) before starting on a proof of the statement that\(\displaystyle (1) \ \ \{ x_\alpha \}_\Delta\) is a basis for \(\displaystyle M \ \Longleftrightarrow \ (3) \ \ M = \bigoplus_{\Delta} x_\alpha R\) with a finite number of elements in \(\displaystyle \Delta\) ... ... (your suggestion of trying with a finite number of elements in \(\displaystyle \Delta\)!)Now ... ... (problem 1!) I am not sure whether Bland means an external or an internal direct sum by \(\displaystyle \bigoplus_{\Delta} x_\alpha R\) ... ... but maybe it does not matter because the internal and external direct sums are isomorphic ... ? (what do you think?)Bland gives the definition of an external direct sum as follows:View attachment 3555Now, we are dealing with a case where\(\displaystyle M = \bigoplus_{\Delta} x_\alpha R \)\(\displaystyle = \{ ( x_\alpha r_\alpha ) \in \prod_\Delta x_\alpha R \ | \ x_\alpha = 0 \text{ for almost all } \alpha \in \Delta \}\)BUT ... we are taking the finite case \(\displaystyle \Delta = \{ x_1, x_2, x_3 \} \)so we can write\(\displaystyle M = \bigoplus_{\Delta} N_\alpha\)\(\displaystyle = \{ ( x_\alpha r_\alpha ) \in N_1 \times N_2 \times N_3 \}\)\(\displaystyle = \{ (x_i r_i ) \ | \ x_i r_i \in N_i \text{ where } i = 1,2,3 \} \)\(\displaystyle = \{ (x_1 r_1, x_2 r_2, x_3 r_3) \ | \ r_1, r_2, r_3 \in R \} \)Now the above seems OK to me ... ... BUT... ... in another post, Euge has indicated that this is incorrect ... ...so it very probably IS incorrect! ... ... but I am not sure why ... ... however, if you can see why it is incorrect, then please let me know so I can proceed with the proof of \(\displaystyle (1) \ \Longleftrightarrow \ (3)\) without delay ... ... Now ... ... suppose that Bland means the internal direct sum by the term \(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\) ... ... then we have the following definition in Bland ... ...View attachment 3556So then, assuming that we have the condition ( ? do we ? ) \(\displaystyle M_\beta \ \bigcap \ \sum_{ \alpha \ne \beta } M_\alpha = 0 \text{ for each } \beta \in \Delta\) , and taking a finite set \(\displaystyle \Delta = \{ x_1, x_2, x_3 \} \),and following the definition,we have:\(\displaystyle M = \bigoplus_{\Delta} x_\alpha R\)\(\displaystyle = x_1R + x_2R + x_3R\)\(\displaystyle = N_1 + N_2 + N_3\), say \(\displaystyle = \sum_\Delta x_i r_i\) since the elements of \(\displaystyle N_i\) are of the form \(\displaystyle x_i r_i\)Now ... which should I assume for the proof of \(\displaystyle (1) \ \Longleftrightarrow \ (3)\) ... ... the external direct sum or the internal direct sum ... ... can you help?Can you also please critique my analysis above? Does it make sense? Are there any erroneous statements?Hope you can help ... ...Peter

I have answered on a PDF. Here's the link:https://www.dropbox.com/s/7dwbjmy6e1e7pna/Peter.pdf?dl=0

 

[Math Amateur]

I have answered on a PDF. Here's the link:https://www.dropbox.com/s/7dwbjmy6e1e7pna/Peter.pdf?dl=0

Hi caffeinemachine,

Thank you so much for your extensive help and guidance ... I really appreciate it!

By the way, you write:

" ... The symbol \(\displaystyle \bigoplus\) always denotes the external direct sum. ..."

and later, you write:

" ... ... BUT I do not appreciate why Bland has, in my opinion rather confusingly, used the notation for external direct sum in place of internal direct sum. ... "Indeed, I think it is somewhat confusing, but it is deliberate ... see the following remark by Bland on page 46:View attachment 3565I think his justification is the isomorphism between the two direct sums as given in his Proposition 2.1.11 on page 48, as follows:https://www.physicsforums.com/attachments/3566Further to your post, I wish say that I found your final remark extremely encouraging ... ... thank you ...

Peter